text:
\section{Introduction\label{sec:intro}}

The present paper continues the study of the asymptotic behaviour of solutions to the conformally invariant wave equation on a Schwarzschild background using analytical and numerical means.  The motivation for this and the previous two articles~\cite{FrauendienerHennig2014,FrauendienerHennig2017} is the desire to better understand the interplay between smoothness of the solutions at null infinity and certain fall-off conditions on their initial data on a Cauchy surface. This is a fascinating property of conformally invariant equations on asymptotically flat spacetimes, which was \JH{discovered by} Friedrich (see~\cite{Friedrich2004} and references therein) in the case of the conformal field equations for gravitational fields.

The situation may be briefly summarised as follows. The idea of using conformal methods for the analysis of asymptotic properties of the gravitational field goes back to Penrose who, in 1965~\cite{Penrose1965}, introduced the concept of asymptotically simple spacetimes as those which admit the attachment of a conformal boundary. In the case of the Minkowski spacetime, the asymptotic boundary turns out to consist of two null hypersurfaces, the past and future null infinities $\mathscr{I}^\pm$. The future cone emanates from a regular point, spacelike infinity $i^0$, and focuses \JH{to} another point, future timelike infinity $i^+$. The past cone behaves similarly, emanating from the past timelike infinity $i^-$ and focusing at $i^0$. Asymptotically flat nonvacuum solutions with vanishing cosmological constant show almost the same structure of two null hypersurfaces as the conformal boundary, except that the ``points'' $i^0$ and $i^\pm$ are no longer regular points. This is not surprising in the case of $i^\pm$, since all the matter in the spacetime must have come out of $i^-$ and will end up in $i^+$. Instead, the singularity of $i^0$ is due to the gravitational field itself. Even in the case of the Schwarzschild spacetime, which does not contain any matter but features a nonvanishing total ADM mass, the ``point'' $i^0$ is singular.

The structure of the conformal boundary suggests that the late incoming radiation and the early outgoing radiation somehow communicate via $i^0$. For a long time, the exact structure of the spacetime region near $i^0$ remained unknown. Several authors tried to capture the essential behaviour by imposing appropriate conditions on the gravitational fields or the metric. But these conditions were to some extent \emph{ad hoc} and did not make full use of the field equations. In~\cite{Friedrich1998}, Friedrich succeeded in shedding more light on spacelike infinity by using all of the available conformal freedom for the Einstein equations. In particular, he used Weyl connections and their associated geodesics to introduce a conformally invariant gauge for the field equations.

When the spacetime structure is analysed using the field equations in this gauge, the previously unstructured singular ``point'' $i^0$ emerges as an entire cylinder $I$ of topology $S^2 \times \mathds R$ meeting $\mathscr{I}^\pm$ in a 2-sphere labelled $I^\pm$. An asymptotically Euclidean hypersurface meets the cylinder in a 2-sphere $I^0$, distinct from $I^\pm$, and initial data for the Einstein evolution induce data for the field variables on $I^0$.

We may think of the cylinder as the bridge across which information travels from $\mathscr{I}^-$ to $\mathscr{I}^+$. This viewpoint is corroborated by the fact that, on the cylinder, the field equations reduce to an entirely intrinsic system of equations. Thus, the cylinder $I$, which is a boundary to the physical manifold, is not a boundary for the field equations in the sense that there is no place to impose boundary conditions on $I$. Indeed, the full set of derivatives of the field variables normal to $I$ is determined completely by an intrinsic symmetric hyperbolic system of transport equations and initial data on $I^0$.

The previously singular nature of $i^0$ is reflected in the fact that the transport equations for the normal derivatives degenerate on the spheres $I^\pm$, where the cylinder meets the null infinities. As a consequence, their solutions develop logarithmic singularities at $I^\pm$ unless their initial data on $I^0$ are restricted appropriately. The structure of the regularity conditions is hierarchical in the sense that the logarithmic divergence in a certain order of the normal derivatives of the field variables can be eliminated provided that they have been eliminated at all lower orders.
Thus, logarithmic divergences up to a certain order can be avoided by suitable change of the initial data subject to fall-off conditions in the physical spacetime.

When the regularity conditions on $I^0$ are violated, there will be singularities at $I^+$. Since $\mathscr{I}^+$ is causally in the domain of influence of $I^+$, the hyperbolic nature of the evolution equations suggests that the singularities will also propagate to null infinity and diminish its smoothness, thus spoiling, in particular, the peeling properties. It is no surprise that a detailed discussion of this behaviour is rather complicated (see the work of Valiente Kroon~\cite{ValienteKroon2012} and references therein) and hence, until now, only some special cases and analogous systems have been analysed in \JH{complete} detail.

Since the degeneracies occur in the part of the conformal field equations addressing the Weyl curvature, namely the Bianchi equations, it is natural to investigate the linearised Bianchi equations. Friedrich~\cite{Friedrich2003} discussed this system on a Minkowski background from the point of view of the well-posedness of the initial value problem. He found estimates for the solutions in terms of the initial data, which show that solutions of these spin-2 zero-rest-mass equations lose smoothness on $\mathscr{I}^+$ depending on how many conditions in the hierarchy of regularity conditions have been satisfied.

Related analytical investigations of the Maxwell equations on a Schwarzschild background were carried out by Valiente Kroon in \cite{ValienteKroon2007,ValienteKroon2008}. It was shown that the electromagnetic field generally contains logarithmic singularities at $I^\pm$, unless the initial data satisfy additional regularity conditions. Moreover, detailed estimates of integrals of the field components and their derivatives in terms of the initial data were derived.

In~\cite{FrauendienerHennig2014} we analysed the behaviour of the conformally invariant wave equation, i.e.\ the zero-rest-mass equation for spin zero, on a Minkowski background,  with spatial infinity represented as the cylinder $I$, in order to find out to what extent the behaviour of the conformal field equations is shared by this simple system. We found that the solutions also need to satisfy certain conditions on $I^0$ to prevent the appearance of logarithmic singularities at $I^+$ (and $I^-$). In~\cite{FrauendienerHennig2017} we generalised this to the conformally invariant wave equation on the Schwarzschild spacetime finding the same general structure, albeit somewhat more complicated: in the Minkowski case, global regularity can be achieved if just a single regularity condition is imposed, which eliminates all singularities. However, in the Schwarzschild case, only rather trivial solutions are regular everywhere.

In both papers \cite{FrauendienerHennig2014,FrauendienerHennig2017}, we restricted ourselves to spherical symmetry to simplify the calculations, and we used fully pseudospectral methods to compute global solutions. Moreover, with our previous investigations in the Schwarzschild case, we could only show existence of logarithmic singularities at $I^+$, but
we were not able to give evidence of the loss of smoothness of the solutions on $\mathscr{I}^+$.
Hence, with the present paper, we pursue two aims: the first is to generalise our study to the \JH{case} without any symmetry, and the second is to use the very direct relationship between the smoothness of functions and \JH{the} convergence properties of their spectral approximations to give clear evidence about the loss of smoothness on $\mathscr{I}^+$ of the solutions depending on how many regularity conditions on $I^0$ have been imposed.

Our main tool for the numerical investigations in this paper is the fully pseudospectral method introduced in \cite{HennigAnsorg2009}, which was first applied to several physical problems in \cite{AnsorgHennig2011} and \cite{Hennig2013}. The basic idea is to approximate the unknown functions using Chebyshev interpolation, and to derive and solve an algebraic system of equations for the Chebyshev coefficients. The algebraic equations are obtained from the requirement that the wave equation or appropriate boundary and regularity conditions are satisfied at a grid of  collocation points. Details can be found in the above references and in \cite{FrauendienerHennig2014, FrauendienerHennig2017}.
The most important advantages of this numerical approach are the extremely accurate solutions that one can typically obtain (usually close to machine accuracy), and the highly-implicit nature of the solution process that effectively couples all gridpoints. A convenient consequence of the latter property is that the Courant-Friedrichs-Lewy (CFL) condition, which restricts the size of the time stepsize in traditional time-marching schemes, is not required here.\footnote{As a related example where the CFL condition creates difficulties, we mention the numerical studies of the spin-2 system on a Minkowski background in \cite{BeyerDoulis2012,BeyerDoulis2014,DoulisFrauendiener2013}. With the numerical method used in these references, it was found that future null infinity $\mathscr{I}^+$ can only be asymptotically approached with a \JH{very large} number of steps. Indeed, the divergence of the characteristic speeds at $\mathscr{I}^+$ enforces, via the CFL condition, \JH{smaller} and smaller time steps. With the fully pseudospectral scheme, on the other hand, we can include $\mathscr{I}^+$ \JH{in} the numerical domain and directly reach it \JH{using only} a moderate number of grid points.}

As discussed above, a very important question that was not addressed in \cite{FrauendienerHennig2017} is whether the logarithmic singularities are restricted to the set $I^+$ or whether they are present on $\mathscr{I}^+$ as well. Here we investigate this question with a careful analysis of the numerical solutions, and we show that singularities indeed also appear everywhere on~$\mathscr{I}^+$.

A particular new feature that we introduce for our numerical considerations in this article is an improved numerical reconstruction of more singular solutions. To this end, we describe a modification of the numerical method, based on a reformulation of the problem in adapted coordinates. This will allow us to resolve solutions with singularities at lower orders with much greater accuracy.

This paper is organised as follows. In Sec.~\ref{sec:coords}, we derive the conformally invariant wave equation and the corresponding decomposition into spherical harmonics in a suitable conformal representation of the Schwarzschild spacetime. \JH{In} Sec.~\ref{sec:cylinder}, we derive analytical properties of the solutions near the cylinder.
In order to test applicability of the pseudospectral scheme to the present problem, we derive simple exact solutions in Sec.~\ref{sec:testsol} that are completely free of singularities,  and we reconstruct them numerically. \JH{In Sec.~\ref{sec:logsing}}, we extend the numerical considerations to singular solutions and localise singularities on $\mathscr{I}^+$. \JH{The reformulation} of the problem for more accurate numerical computations  \JH{is} described in Sec.~\ref{sec:reg}. Finally, in Sec.~\ref{sec:discussion}, we summarise and discuss our results.

\section{Coordinates and wave equation\label{sec:coords}}

We study the conformally invariant wave equation for a scalar function $f$, i.e.\ the equation
\begin{equation}\label{eq:CWE}
 g^{ab}\nabla_a\nabla_b f-\frac{R}{6}f=0.
\end{equation}
Here, $g$ is a fixed background metric, which we take to be conformal to the Schwarzschild metric, and $R$ is the corresponding scalar curvature. We want to consider this equation near the cylinder representation $I$ of spacelike infinity $i^0$ and near a portion of future null infinity $\mathscr{I}^+$. To this end,  we use the same coordinates $(\tau,\rho,\theta,\phi)$, first introduced by Friedrich \cite{Friedrich2004}, that we also utilised in \cite{FrauendienerHennig2017}. In terms of these coordinates, the conformal metric takes the form
\begin{equation}\label{eq:conmet}
 g=\frac{2}{\rho}A\,\mathrm{d}\rho\,\mathrm{d}\tau-\frac{1-\tau}{\rho^2}A[2-(1-\tau)A]\mathrm{d}\rho^2-\mathrm{d}\theta^2-\sin^2\theta\,\mathrm{d}\phi^2,
\end{equation}
where
\begin{equation}\label{eq:defA}
 A:=\frac{(1-r)(1+\rho)^3}{(1-\rho)(1+r)^3},\quad r:=\rho(1-\tau).
\end{equation}
This is related to the physical metric $\tilde g$ of the Schwarzschild spacetime with mass parameter $M$ via $g=\Theta^2\tilde g$ with conformal factor
\begin{equation}\label{eq:Theta}
 \Theta=\frac{2r}{M(1+r)^2}.
\end{equation}
We consider the coordinates in the intervals $0\le\rho\le\rho_\mathrm{max}<1$ and $0\le\tau\le1$, where $\rho=0$ corresponds to the (future half of the) cylinder $I$ and $\tau=1$ to a part of $\mathscr{I}^+$. The coordinates are adapted to one family of radial null geodesics, which take the simple form $\rho=\mathrm{constant}$. Note that these coordinates only cover a part of the extended Schwarzschild spacetime, and, in particular, do not extend to the event horizon, cf.~\cite{FrauendienerHennig2017} for more details. Here, however, we are mainly interested in a vicinity of infinity. Nevertheless, if required, the coordinates can be modified to cover a larger region including the horizon \cite{FrauendienerHennig2017}.

In terms of the background metric \eref{eq:conmet}, the conformally invariant wave equation for a function $f=f(\tau,\rho,\theta,\phi)$ reads
\begin{equation}\fl
 (1-\tau)[2-(1-\tau)A] f_{,\tau\tau}+2\rho f_{,\tau\rho}
 -2\left[1-(1-\tau)A\frac{1-2r}{1-r^2}\right]f_{,\tau}
 -A\bigtriangleup_\Omega f+\frac{4rA}{(1+r)^2}f=0,
\end{equation}
where $\bigtriangleup_\Omega f=\frac{1}{\sin\theta}(\sin\theta\,f_{,\theta})_{,\theta}-\frac{1}{\sin^2\theta} f_{,\phi\phi}$ is the angular part of the flat Laplacian. Since the $\theta$- and $\phi$-derivatives only appear in the form of $\bigtriangleup_\Omega$, we can decompose the solutions into spherical harmonics $Y_{lm}$ in the usual way,
\begin{equation}
 f(\tau,\rho,\theta,\phi)=\sum_{l=0}^\infty\sum_{m=-l}^l\psi_{lm}(\tau,\rho)Y_{lm}(\theta,\phi).
\end{equation}
This reduces the wave equation to a set of independent equations for the modes $\psi_{lm}$,
\begin{eqnarray}\label{eq:modes}
 (1-\tau)[2-(1-\tau)A] \psi_{lm,\tau\tau}+2\rho \psi_{lm,\tau\rho}
 -2\left[1-(1-\tau)A\frac{1-2r}{1-r^2}\right]\psi_{lm,\tau} \nonumber\\
 \quad +Al(l+1) \psi_{lm}+\frac{4rA}{(1+r)^2}\psi_{lm}=0.
\end{eqnarray}
Note that only $l$ but not $m$ appears in the equations. Effectively, we can therefore study $1+1$ dimensional problems for each $l$. For ease of notation, we will usually assume that we have made a fixed choice of $l$ (and $m$) and denote the corresponding mode simply by $\psi$.

\section{Behaviour near the cylinder\label{sec:cylinder}}

In the spherically symmetric case, discussed in \cite{FrauendienerHennig2017}, we found that the wave function and its spatial derivatives at the cylinder $\rho=0$ generally have logarithmic terms. As a consequence, certain mixed derivatives diverge as $\tau\to1$, i.e.\ at the set\footnote{Note that $I^+$ is a topological sphere, but in two dimensional diagrams that suppress the angular coordinates, it is represented by a point.} $I^+$. The order at which the first singularities appear depends on the choice of initial data, and we can systematically construct data for which the lowest singularities up to a certain order are all eliminated. In the following we show that these results carry over to the nonspherical case.

We consider a solution $\psi(\tau,\rho)$ that can be expanded in a Taylor series about $\rho=0$ with $\tau$-dependent coefficients,
\begin{equation}\label{eq:expan}
 \psi(\tau,\rho)=\psi_0(\tau)+\rho\psi_1(\tau)+\rho^2\psi_2(\tau)+\dots
\end{equation}
Plugging this into \eref{eq:modes}, we obtain a system of ODEs of the form
\begin{equation}\label{eq:ODE}
 (1-\tau^2)\ddot\psi_n+2(n-\tau)\dot\psi_n+l(l+1)\psi_n=R_n,
\end{equation}
where the dot denotes differentiation with respect to $\tau$.
The inhomogeneity $R_n$ on the right-hand side depends linearly on up to six previous functions $\psi_{n-1}$, $\psi_{n-2},\dots$ and their first and second order $\tau$-derivatives. While $R_n$ can be given explicitly as a lengthy expression, we can study some properties of the solutions without using this explicit form.

In the spherically symmetric case $l=0$, we can easily integrate \eref{eq:ODE} directly. For $l>0$, we can solve \eref{eq:ODE} by observing that an appropriate rescaling of $\psi_n$ transforms the equation into an inhomogeneous associated Legendre equation. It turns out that  we have to dicsuss the two cases $n\le l$ and $n>l$ separately.

We begin with the case $n=0,1,\dots,l$. If we abbreviate the associated Legendre functions of the first and second kind with $u=P^n_l(\tau)$ and $v=Q^n_l(\tau)$, then the general solution to \eref{eq:ODE} can be written as
\begin{eqnarray}\label{eq:ODEsol1}\
 \psi_n(\tau)&=&\frac{(l-n)!}{(l+n)!} \alpha_n(\tau)
 \left[\left(-\int_0^\tau\frac{R_n(\tau') v(\tau')}{\alpha_n(\tau')}\,\mathrm{d}\tau'+c_1\right)u(\tau)\right.\nonumber\\
 &&\qquad
      +\left.\left(\int_0^\tau\frac{R_n(\tau') u(\tau')}{\alpha_n(\tau')}\,\mathrm{d}\tau'+c_2\right)v(\tau)\right]
\end{eqnarray}
in terms of two integration constants $c_1$, $c_2$ and with
\begin{equation}
 \alpha_n(\tau):=\left(\frac{1-\tau}{1+\tau}\right)^{\frac n2}.
\end{equation}
From this result we can now derive the following. If we assume that, for fixed $l$, $R_n$ is regular (namely if singularities at previous orders $n-1$, $n-2,\dots$ have all been eliminated), then it is possible to choose a specific value for $c_2$ (and an arbitrary value for $c_1$) such that the solution $\psi_n$ is regular as well. More explicitly, using certain properties of associated Legendre functions, the solution \eref{eq:ODEsol1} can be reformulated as
\begin{eqnarray}\label{eq:ODEsol2}
 \psi_n &=& \frac{(l-n)!}{2(l+n)!}P^{(n)}_l
  \Bigg[\sum_{k=0}^{n-1}\Bigg(\begin{array}{c}n\\k\end{array}\Bigg)
   (-1)^{n-k}\frac{\mathrm{d}^{n-k-1}}{\mathrm{d}\tau^{n-k-1}}\left(\tilde R P^{(k)}_l\right)\Big|_{\tau=1}\nonumber\\
  && +\int_0^1\tilde R P^{(n)}_l\mathrm{d}\tau-(-1)^n c_2\Bigg](1-\tau)^n\ln(1-\tau)
  +\textrm{regular terms},
\end{eqnarray}
where we abbreviated the $k$th derivative of the Legendre polynomial $P_l$ with $P^{(k)}_l$, and where we defined $\tilde R:=(1+\tau)^n R_n$. Hence we observe that, if previous orders are free of singularities, then the function $\psi_n$ generally contains a logarithmic term of the form $(1-\tau)^n\ln(1-\tau)$. However, we can always choose $c_2$ such that the square bracket in \eref{eq:ODEsol2} vanishes, corresponding to a particular choice of initial data. In that case $\psi_n$ is regular too. Therefore, by imposing more and more constraints onto the initial data, we can step-by-step eliminate more and more singular terms.

Now we look at the second case $n=l+1,l+2,\dots$ For $n>l$, the above function $u=P_l^n(\tau)$ vanishes identically, because it is proportional to the $(n+l)$th derivative of a polynomial of degree $2l$, namely to $\frac{\mathrm{d}^{n+l}}{\mathrm{d}\tau^{n+1}}(1-\tau^2)^l$. Consequently, Eq.~\eref{eq:ODEsol1} does not provide the general solution in that case. We find, however, that the replacement
\begin{equation}
 \frac{\mathrm{d}^{n+l}}{\mathrm{d}\tau^{n+l}}(1-\tau^2)^l\to
 \lim_{\varepsilon\to0}\frac{1}{\varepsilon}\frac{\mathrm{d}^{n+l}}{\mathrm{d}\tau^{n+l}}(1-\tau^2)^{l+\varepsilon}
 =\frac{\mathrm{d}^{n+l}}{\mathrm{d}\tau^{n+l}}(1-\tau^2)^l\ln(1-\tau^2)
\end{equation}
does not vanish for $n>l$, but still leads to a solution. Based on this observation, we can construct the general solution in terms of the new functions
\begin{equation}
 u:=(1-\tau)^n\frac{\mathrm{d}^n}{\mathrm{d}\tau^n}[P_l\ln(1-\tau^2)],\quad
 v:=(1-\tau)^n\frac{\mathrm{d}^n}{\mathrm{d}\tau^n}\left[P_l\ln\frac{1-\tau}{1+\tau}\right].
\end{equation}
Note that the Wronskian $W$ of $u$ and $v$ turns out to be of the form
$W\equiv u\dot v-\dot u v=c(1-\tau)^{n-1}/(1+\tau)^{n+1}$. In terms of the constant $c$ in this formula, the general solution can be written as
\begin{equation}\label{eq:ODEsol3}
 \fl
 \psi_n=\frac{1}{c}\left[\left(-\int_0^\tau\left(\frac{1+\tau'}{1-\tau'}\right)^nR_n v\,\mathrm{d}\tau'+c_1\right)u
 +\left(\int_0^\tau\left(\frac{1+\tau'}{1-\tau'}\right)^nR_n u\,\mathrm{d}\tau'+c_2\right)v\right].
\end{equation}
With a similar reformulation as above, we find that, for regular $R_n$, the particular solution with $c_1=c_2=0$ generally behaves like $(1-\tau)^n\ln(1-\tau)$ near $\tau=1$. Here, however, it is not possible to choose $c_1$ and $c_2$ in the general solution such that this singularity is eliminated, because $u$ and $v$, and, therefore, any linear combinations $c_1u+c_2v$ are regular functions for $n>l$. Consequently, they cannot compensate for singularities in the integral terms. Nevertheless, we can still eliminate the logarithmic singularity in $\psi_n$ by appropriately choosing integration constants that have been introduced at \emph{previous} orders (i.e.\ in the solutions for  $\psi_{n-1}$, $\psi_{n-2}, \dots$). The values of those earlier integration constants change the form of the inhomogeneity $R_n$ and, therefore, the integrals in \eref{eq:ODEsol3}.

In order to illustrate the above considerations, we look in more detail at the cases $l=0,\dots, 4$ and $n=0,\dots,4$. In each case, we translate the condition for regularity on the integration constants into a condition for the initial data. It turns out that we have to choose data $\psi(\tau=0,\rho)$ and $\psi_{,\tau}(\tau=0,\rho)$ for which certain linear combinations of these functions and their $\rho$-derivatives vanish at $\rho=0$. The resulting conditions are listed in Table \ref{tab}. If, for a fixed $l$, all conditions up to a certain $n$ are satisfied, then all functions $\psi_0,\dots,\psi_n$ will be regular at $\tau=1$. Hence, the corresponding logarithmic terms do not appear for data subject to these conditions. Note that the regularity conditions with $l=0$ and $n=0,1,2,3$ were already derived in \cite{FrauendienerHennig2017}.

\begin{table}
{\small\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|p{2mm}||p{2.3cm}|p{2.2cm}|p{2.5cm}|p{2.96cm}|p{3.3cm}|}
\hline
 $n$ & $l=0$ & $l=1$ & $l=2$ & $l=3$ & $l=4$ \\
\hline\hline
 $0$ & $\psi_{,\tau}=0$ & $\psi=0$ & $\psi_{,\tau}=0$ & $\psi=0$ & $\psi_{,\tau}=0$\\
\hline
 $1$ & NA
     & $\psi_{,\tau\rho} + 10 \psi_{,\tau}\newline + \psi_{,\rho} = 0$
     & $\psi_{,\rho} + 18 \psi = 0$
     & $\psi_{,\tau\rho}+\frac{70}{3} \psi_{,\tau}\newline +\psi_{,\rho}=0$
     & $9 \psi_{,\rho}+274 \psi=0$\\
\hline
 $2$ & $\psi_{,\tau\rho}+\psi_{,\rho}\newline =0$
     & NA
     & $\psi_{,\tau\rho\rho}+2 \psi_{,\rho\rho}\newline
       +\frac43 (23 \psi_{,\tau\rho}\newline -\frac{9434}{35} \psi)=0$
     & $945 \psi_{,\rho\rho}\newline +51660 \psi_{,\rho}\newline-43264 \psi_{,\tau}=0$
     & $\psi_{,\tau\rho\rho}+2 \psi_{,\rho\rho}\newline
      +\frac85 (39 \psi_{,\tau\rho}\newline
      -\frac{452902}{693}\psi)=0$\\
\hline
 $3$ & $3 \psi_{,\rho\rho} +14 \psi_{,\rho} \newline -36 \psi=0$
     & $\psi_{,\tau\rho\rho}+2 \psi_{,\rho\rho}\newline -36 \psi_{,\tau}=0$
     & NA
     & $\frac{105}{4} (\psi_{,\tau\rho\rho\rho
       +3 \psi_{,\rho\rho\rho}\newline
       +\frac{294}{5} \psi_{,\tau\rho\rho})\newline
       +147124 \psi_{,\tau\rho}\newline
       +(\frac{46785491}{15}\newline -173056\ln2) \psi_{,\tau}\newline =0$
     & $\frac{1}{192} \psi_{,\rho\rho\rho
      +\frac{11}{20} \psi_{,\rho\rho}\newline
      -\frac{2312}{3465} \psi_{,\tau\rho}\newline
      -(\frac{143840317}{498960}\newline
      +\frac{73984}{693}\ln2)\psi=0$\\
\hline
 $4$ & $\psi_{,\tau\rho\rho\rho}+3 \psi_{,\rho\rho\rho}\newline
       +45 \psi_{,\tau\rho\rho} \newline
       +8 (77 \psi_{,\tau\rho}\newline +184 \psi)=0$
     & $10 \psi_{,\rho\rho\rho}\newline +123 \psi_{,\rho\rho}\newline
       +\frac{32}{5} (361\psi_{,\tau}\newline
       -90\psi_{,\rho})=0$
     & $\psi_{,\tau\rho\rho\rho}+3 \psi_{,\rho\rho\rho}\newline
      -\frac{2}{35}(3045 \psi_{,\tau\rho}\newline -34586 \psi)=0$
     & NA
     & $\frac18 \psi_{,\tau\rho\rho\rho\rho}+\frac12 \psi_{,\rho\rho\rho\rho}\newline
        +\frac{407}{35}\psi_{,\tau\rho\rho\rho}\newline
        +24(\frac{407}{5}\ln2\newline
        -\frac{633291983}{3203200}) \psi_{,\rho\rho}\newline
        -96(\frac{407}{5}\ln^22\newline
        +\frac{1066985657107}{6054048000}\newline
        -\frac{157035547}{2882880}\ln2) \psi_{,\tau\rho}\newline
        +256(\frac{396300733}{320320}\ln^22\newline
        +\frac{17520976798265933}{1598268672000}\newline
        -\frac{452636099453}{271814400}\ln2\newline
        -\frac{10693}{126}\pi^2\newline
        -814\ln^32) \psi=0$\\
\hline
\end{tabular}
}
\caption{Regularity conditions for the initial data that must be satisfied at $\rho=\tau=0$ for various values of $n$ and $l$. Note that ``NA'' indicates that the corresponding  function $\psi_n$ is automatically regular so that no condition needs to be imposed to achieve regularity at that order.\label{tab}}
\end{table}

\section{Simple exact solutions\label{sec:testsol}}

As a first test of the performance of the pseudospectral numerical procedure, when applied to the nonspherical wave equation, we construct simple exact ``test solutions''. Then we can  try to reconstruct them numerically. To this end, we follow the approach from \cite{FrauendienerHennig2017} and consider the conformally invariant wave equation with respect to the physical metric $\tilde g$ in (isotropic) Schwarzschild coordinates. By restricting ourselves to time-independent modes $\tilde\psi$, we can reduce the equation to an ODE. A solution $\tilde\psi$ can then be transformed to a solution $\psi$ with respect to our conformally rescaled metric \eref{eq:conmet} using $\psi=\Theta^{-1}\tilde\psi$ (since the conformal weight in four spacetime dimensions is $-1$).

The ODE in question can be written as
\begin{equation}
 \frac{r^2}{1-r^2}\left((1-r^2)\tilde\psi_{,r}\right)_{,r}-l(l+1)\tilde\psi=0.
\end{equation}
The general solution can be given in terms of hypergeometric functions and two integration constants. In order to achieve regularity at the cylinder (i.e.\ at $r=0$), one of the integration constants must be chosen appropriately, while the other constant can have an arbitrary value (and the particular value is not important as it simply rescales the solution by multiplication with a constant factor).
Using $r=\rho(1-\tau)$, the solution obtains a nontrivial time dependence with respect to our time coordinate $\tau$.

We explicitly construct the test solutions for $l=0,\dots, 4$. After a certain choice of the integration constants, these are
\begin{eqnarray}
 l=0:\quad && \psi=\frac{(1+r)^2}{r}\ln\frac{1+r}{1-r},\\
 l=1: && \psi=\frac{(1+r)^2}{r^2}\left((1+r^2)\ln\frac{1+r}{1-r}-2r\right),\label{eq:l=1test}\\
 l=2: && \psi=\frac{(1+r)^2}{r^3}\left((3r^4+2r^2+3)\ln\frac{1+r}{1-r}-6r(1+r^2)\right),\\
 l=3: && \psi=\frac{(1+r)^2}{r^4}\Bigg(3(1+r^2)(5r^4-2r^2+5)\ln\frac{1+r}{1-r}
         \nonumber\\
      && \qquad\qquad -2r(15r^4+14r^2+15)\Bigg),\\
 l=4: && \psi=\frac{(1+r)^2}{r^5}\Bigg((105r^8+60r^6+54r^4+60r^2+105)\ln\frac{1+r}{1-r}
         \nonumber\\
      && \qquad\qquad -10r(1+r^2)(21r^4-2r^2+21)\Bigg).
\end{eqnarray}
Note that the terms in large parentheses all vanish sufficiently quickly as $r\to0$ and  compensate the singularities of the first factors at $r=0$. In fact, the solutions are analytic in our domain $0\le\tau\le1$, $0\le\rho\le\rho_\mathrm{max}<1$.

Now we prescribe initial data for these test solutions at $\tau=0$ and see how accurately we can recover the corresponding solutions numerically. \JH{An} interesting numerical detail is as follows. In the limit $\tau\to1$, the wave equation \eref{eq:modes} becomes
\begin{equation}\label{eq:atScri}
 2(\rho\psi_{,\tau\rho}-\psi_{,\tau})-l(l+1)\psi=0.
\end{equation}
In the spherically symmetric case $l=0$, this is an intrinsic equation for $\psi_{,\tau}$ with solution $\psi_{,\tau}=c\rho$. The integration constant $c$, however, cannot be determined from the wave equation at $\mathscr{I}^+$. Consequently, we observed in \cite{FrauendienerHennig2017} that we cannot just impose the wave equation itself at all grid points at $\mathscr{I}^+$; an extra condition is required that fixes the integration constant. This can be done by prescribing the exact value of the mixed derivative $\psi_{,\tau\rho}$ at the single point $\rho=0$, $\tau=1$, which equals $c$. This value can easily be obtained from the analysis at the cylinder.

Is such an extra condition also required for nonspherical waves, i.e.\ for $l>0$? In that case, \eref{eq:atScri} is no longer an intrinsic equation for $\psi_{,\tau}$, due to the additional $l(l+1)$-term. Hence it should be possible to impose the wave equation at all grid points. On the other hand, if extra information about the solution to a PDE problem is available and can explicitly be enforced in the numerical code, this may improve the numerical accuracy. \JH{Therefore, it is worth comparing} the numerical performance with and without an extra condition at $\rho=0$, $\tau=1$. As an extra condition, we can again give the exact value for $\psi_{,\tau\rho}$. For example, in the case $l=1$, this condition reads
\begin{equation}\label{eq:extra}
 l=1:\quad \psi_{,\tau\rho}\big|_{\tau=1,\rho=0}
     = - 4(1-\ln 2) \psi_{,\tau} - \psi_{,\rho}\big|_{\tau=0,\rho=0},
\end{equation}
where the right-hand side can be computed from the initial data at $\tau=0$.

\begin{figure}\centering
 \includegraphics[width=8cm]{ExtraConditionX.pdf}
 \caption{The error in the numerical reconstruction of the $l=1$ test solution \eref{eq:l=1test}  with parameter $\rho_\mathrm{max}=0.5$. The resolutions (i.e.\ number of collocation points) in spatial and time direction are denoted with $n_\sigma$ and $n_\tau$, respectively. In all examples discussed in this article we always choose $n_\sigma=n_\tau$, but other choices are of course possible as well.\label{fig:extra}}
\end{figure}
The corresponding convergence plot is shown in Fig.~\ref{fig:extra}. As a measure \JH{of} the numerical error, we compute the largest absolute difference between the exact and numerical solutions at $100\times100$ equidistant grid points. As expected, we find that the solution can be obtained with good accuracy by just imposing the wave equation at all grid points. However, the accuracy significantly further improves if we use the extra condition \eref{eq:extra}. For comparison, \JH{Fig.~\ref{fig:extra}} also shows the Chebyshev interpolation error, which is the largest difference between the exact solution and a Chebyshev approximation to this solution (without solving any differential equation). We see that the extra condition allows the numerical accuracy to reach this ``theoretical threshold.''

For the test solutions with other values of $l$, we qualitatively obtain the same picture. Hence, from now on, we will always impose the extra conditions at $\rho=0$, $\tau=1$, in order to optimise \JH{numerical accuracy.}

Another important feature of the convergence plot in Fig.~\ref{fig:extra} (and the plots obtained for other $l$ values) is that the error approximately follows a straight line \JH{on a logarithmic scale}, until the accuracy is close to machine accuracy. This is exactly what is expected for sufficiently regular (namely $C^\infty$ or analytic) solutions like our test solutions.

\section{Logarithmic singularities\label{sec:logsing}}

We have seen that the numerical method enables us to very accurately reconstruct the analytic test solutions. For general initial data, however, we know that logarithmic singularities form on the cylinder. Moreover, we should expect that these singularities also ``spread'' to $\mathscr{I}^+$. Whether this is true will be investigated in the second half of this section.

Similarly to our considerations in \cite{FrauendienerHennig2017}, we start by constructing families of initial data that have their leading singularities on the cylinder at a prescribed order. For the first few values of $l$, we choose the following data at $\tau=0$, which depend on two free parameters $\alpha$, $\beta$,
\begin{eqnarray}\label{eq:ab1}
 \fl l=1:&& \psi = \rho-\frac{288+\beta}{2065}\sin(5\rho),\quad
  \psi_{,\tau} = \frac{\alpha}{10}\cos(2\rho)-\left(\alpha+\frac{125-\beta}{413}\right)\rho,\\
 \fl l=2: && \psi=\rho-\frac15\sin(5\rho),\nonumber\\
 \fl &&\psi_{,\tau}=\frac{3}{184}\left(\frac12(87\alpha+\beta+1150)\sin(2\rho)-(89\alpha+\beta+1150)\rho\right),\\
 \fl l=3: && \psi = \rho+\big[865280(10816\alpha-12915)\ln2+17307471168\alpha -54080\beta
    \nonumber\\
  \fl&&\qquad -28622700215\big]/[25(2235018240\ln2+5745834043)]\sin(5\rho),\nonumber\\
 \fl  && \psi_{,\tau}=
 \big[15(-6450348736\alpha-51660\beta+1017056256)\cos(2\rho)\nonumber\\
 \label{eq:ab3}
 \fl &&\qquad -112306421760(\ln2)\alpha\rho+(2049932403584\alpha+18729960\beta\nonumber\\
 \fl &&\qquad -36874608750)\rho\big]/[134101094400\ln2+344750042580].
\end{eqnarray}
These families are constructed, using the regularity conditions in Table~\ref{tab}, such that for any choice of the parameters, the functions $\psi_0$ and $\psi_1$ in the expansion \eref{eq:expan} are free of logarithmic terms.\footnote{This is the minimum requirement to ensure that the wave function $\psi$ and the derivatives up to second order, i.e.\ the quantities that explicitly appear in the wave equation, are bounded in our numerical domain. Otherwise, our numerical method fails and cannot solve the problem. However, we will later consider the equation in different coordinates that allow us to reconstruct even such ``very singular'' solutions, see Sec.~\ref{sec:reg} below.} Moreover, for $\alpha\neq0$, the next possible logarithmic singularity is present at the cylinder (for $l=1$ a singularity in the function $\psi_3$, since $\psi_2$ is then automatically regular; and a singularity in $\psi_2$ for $l=2, 3$). This singularity disappears for $\alpha=0$. For $\beta=0$, the following logarithmic term vanishes as well.

\begin{figure}\centering
 \includegraphics[width=7.6cm]{alpha-beta1X.pdf}\quad
 \includegraphics[width=7.6cm]{alpha-beta2X.pdf}\\
 \includegraphics[width=7.6cm]{alpha-beta3X.pdf}
 \caption{Convergence plots for the families of initial data \eref{eq:ab1}--\eref{eq:ab3}  for various parameter values in the cases $l=1, 2, 3$ (and with coordinate parameter $\rho_\mathrm{max}=0.5$).\label{fig:alphabeta}}
\end{figure}
We solve the corresponding initial value problems for different values of the parameters, in order to compare examples in which the leading logarithmic terms appear at different orders and have coefficients of different magnitudes. The numerical accuracy is then tested by estimating the relative error of the function value $\psi$ at $\tau=1$, $\rho=\rho_\mathrm{max}$ (i.e.\ a point at $\mathscr{I}^+$), which is done by comparing the function values as obtained with different resolutions with the value corresponding to the highest resolution of $n_\sigma=n_\tau=52$. The resulting convergence plots are shown in Fig.~\ref{fig:alphabeta}.

We observe that the accuracy generally increases for decreasing values of the parameters $\alpha$ and $\beta$, since the coefficients of the leading singular terms then become smaller or even vanish (if $\alpha=0$ or $\beta=0$ are chosen). Clearly, more regular solutions can be resolved with greater accuracy. Most importantly, the curves roughly follow straight lines (for sufficiently large resolutions) \JH{on} a \emph{double}-logarithmic scale, corresponding to \emph{algebraic} rather than exponential convergence. This shows that these solutions only have finite differentiability $C^k$ instead of being $C^\infty$. In such cases the error is indeed expected to behave like a negative power of the resolution, which is exactly the observed algebraic convergence. Evidently, the logarithmic singularities have a strong influence on the numerical results. In particular, if logarithmic terms are already present at low orders, then much larger resolution \JH{is} required --- compared to the previously discussed analytic test solutions --- to obtain a reasonable accuracy. This is qualitatively the same behaviour that we already found for $l=0$ in \cite{FrauendienerHennig2017}.

An important question is whether the singularities are all concentrated at $I^+$ (i.e.\ at $\tau=1$, $\rho=0$), where the cylinder and $\mathscr{I}^+$ approach each other, or whether the solutions also have a limited differentiability as $\tau\to1$ for $\rho>0$. This will be investigated in the following.

In order to study the regularity of a solution $\psi$ as $\mathscr{I}^+$ is approached, we  consider $\psi(\tau,\rho)$ as a 1-dimensional function of $\tau$ for some fixed $\rho$. From the numerical solution we read off the function values of $\psi$ at the collocation points on this line (via Chebyshev interpolation). Then we compute the Chebyshev approximation of this 1-dimensional function. The fall-off behaviour of the Chebyshev coefficients $c_k$ depends on the regularity of the function: for a $C^\infty$ function, the coefficients fall off exponentially, while for a $C^k$ function, they fall off algebraically. In particular, if a function behaves like $(1-\tau)^s\ln(1-\tau)$ (plus weaker singular and plus regular terms) near $\tau=1$ for some integer $s$ and is $C^\infty$ for $0\le\tau<1$, then, for sufficiently large $k$, we expect\footnote{This can be shown with repeated partial integration of the integral formula for the Chebyshev coefficients.} $c_k\sim k^{-(2s+1)}$. If we plot $\log_{10}|c_k|$ versus $\log_{10}k$ in a double-logarithmic diagram, then we should roughly find a straight line with slope $-(2s+1)$, at least in an appropriate range of $k$ values.

\begin{figure}\centering
 \includegraphics[width=7.6cm]{singularities1X.pdf}\quad
 \includegraphics[width=7.6cm]{singularities2X.pdf}
 \caption{Fall-off of the Chebyshev coefficients $c_k$ of $\psi(\tau,\rho)$, considered as a 1-dimensional function of $\tau$ for different fixed values of $\rho$. The solutions correspond to the case $l=1$ with initial data \eref{eq:ab1} with parameters $\alpha=\beta=1$ (left panel) and $\alpha=0$, $\beta=1$ (right panel). For comparison, dotted straight lines with slopes $-7$ (left panel) and $-11$ (right panel) are shown. Numerical parameters: $\rho_\mathrm{max}=0.5$, $n_\sigma=n_\tau=50$.\label{fig:sing}}
\end{figure}
Figure~\ref{fig:sing} illustrates the observed fall-off for two solutions, corresponding to different initial data in the case $l=1$. In the first case (as shown in the left panel), we approximately find straight lines with slope $-7$ for sufficiently large $k$ (but $k$ not too close to the maximum\footnote{If $k$ is close to the number of collocation points (here $50$, i.e.\ for $\log_{10}k$ close to $1.7$), then we see deviations from the linear behaviour. This comes from a combination of two effects. The first is due to numerical errors, which lead to the strongest deviations from the straight line for the smallest Chebyshev coefficients\JH{, with magnitude of} the order of the numerical error. The second effect comes from the difference between the coefficients of an infinite Chebyshev series (which strictly have the algebraic fall-off discussed above) and the coefficients in Chebyshev interpolation with a finite number of coefficients (as considered here), which becomes most significant if $k$ is close to the maximum, see~\cite{Boyd}.}). For different values of $\rho$, the magnitudes of the Chebyshev coefficients change (and they tend to become smaller as $\rho\to0$, in accordance with regularity of $\psi$ on the cylinder $\rho=0$, where only $\rho$-derivatives of $\psi$ have logarithmic terms, but not $\psi$ itself), but they approach zero at the same rate as the resolution increases. This is consistent with a logarithmic singularity of the form $(1-\tau)^3\ln(1-\tau)$. In the second case (right panel), we observe a slope of about $-11$, corresponding to a logarithmic term $(1-\tau)^5\ln(1-\tau)$.

In addition to the examples from Fig.~\ref{fig:sing}, we have investigated the fall-off for many other solutions. In particular, for all slopes in the set $\{-5,-7,-9,-11,-13\}$, we found examples for which the Chebyshev coefficients have a fall-off corresponding to straight lines with these slopes. This is consistent with leading logarithmic terms of the form $(1-\tau)^s\ln(1-\tau)$ with $s=2,3,4,5,6$. Of course, these observations are no rigorous proof of exactly that form of singularity, as other types of singularities could lead to a similar fall-off behaviour. But inspired \JH{by} our analytical considerations on the cylinder, we strongly expect to find exactly those logarithmic singularities. Depending on the initial data, the leading-order singularity can appear at different orders, as our examples show.

One important point that still needs some attention is to verify whether the singularities are indeed concentrated at $\mathscr{I}^+$. In other words, we have to make sure that along the considered lines $\rho=\mathrm{constant}$ no singularities occur before $\tau=1$ is reached. This is already clear from an analytical point of view, since solutions to the \emph{linear} wave equation on a Schwarzschild background corresponding to smooth initial data are smooth. But we want to verify that the same shows in our numerics.
To this end, we can use the same approach as before, but this time applied to lines $\tau=\mathrm{constant}$. If $\psi$ is regular on such lines for $\tau<1$, then the Chebyshev coefficients of $\psi$ (now taken to be a function of $\rho$) should fall off exponentially, rather than algebraically. Corresponding numerical experiments indeed verify that this is the case.

Another way to show that the region $\tau<1$ is free of singularities is to solve the wave equation in the restricted domain $0\le\rho\le\rho_\mathrm{max}$, $0\le\tau\le\tau_\mathrm{max}$ for some $\tau_\mathrm{max}<1$ but close to $1$. If there are no singularities in this domain, then the measured numerical error should decrease exponentially. This is exactly what we observe, see Fig.~\ref{fig:smaller}.
\begin{figure}\centering
 \includegraphics[width=8cm]{SmallerTimeIntervalX.pdf}
 \caption{Convergence plot for two solutions corresponding to the same sets of initial data as considered in Fig.~\ref{fig:sing}, but this time computed in the smaller time interval $0\le\tau\le0.95$. The relative error quickly reaches values below $10^{-13}$, and the observed exponential convergence shows that this domain is free of singularities.\label{fig:smaller}}
\end{figure}

We therefore conclude that singularities occur at $\tau=1$ for all $\rho$-values, i.e.\ not just at the cylinder but at $\mathscr{I}^+$ as well, while the solutions are $C^\infty$ in the region $\tau<1$. The observed fall-off of the Chebyshev coefficients is consistent with (\JH{though not rigorous proof of) the} presence of logarithmic singularities at certain leading orders. The exact order depends on the particular solution, i.e.\ on the choice of initial data.

\section{Regularisation of (too) singular solutions\label{sec:reg}}

The previous numerical calculations show that the technique of fully pseudospectral time evolution can successfully be applied to nonspherical modes of the Schwarzschild wave equation. We also observe that the numerical accuracy depends on the lowest order at which logarithmic singularities occur. If this order is not ``too low'', then highly accurate solutions are possible. Otherwise, however, only a moderate accuracy can be expected. If the solutions are singular enough so that even derivatives of second order or lower diverge as $\tau\to 1$, then quantities appearing in the equation itself are unbounded. In this case no reliable degree of accuracy should be expected at all.

This raises the question as to whether the formulation of the problem can be changed to allow for more accurate numerical solutions and to make even initial value problems with  very singular solutions tractable. Based on our observation that logarithmic singularities appear at $\mathscr{I}^+$, a possible approach is to reformulate the wave equation in terms of different coordinates such that $\psi$ as a function of these new coordinates is more regular. This will be described in the following.

We use a generalisation of a coordinate transformation that was applied in \cite{KalischAnsorg} to construct black string solutions in higher dimensions.\footnote{Note that the problem studied in \cite{KalischAnsorg} required the solution of elliptic equations to construct a stationary equilibrium configuration. At a certain spatial boundary, logarithmic singularities were observed and regularised with a spatial coordinate transformation. Here we apply a generalisation of the same transformation to the \emph{time} coordinate, which turns out to work as well. An important new ingredient provided here is the free parameter $w$ in \eref{eq:trans}.} To this end, we replace our time coordinate $\tau$ with a new coordinate $t$ by setting
\begin{equation}\label{eq:trans}
 \tau=1-\mathrm{e}^{-\frac{wt}{1-t}},
\end{equation}
where $w>0$ is a parameter. Suitable values for $w$ have to be identified with numerical experiments, and we will see that an appropriate choice of $w$ is crucial for \JH{numerical accuracy.} Note that this transformation maps the interval $[0,1]$ to itself. Moreover, a term of the form $(1-\tau)^{s_1}\ln^{s_2}(1-\tau)$ with $s_1,s_2>0$
(which is a slightly more general logarithmic singularity with respect to $\tau$ than discussed before), becomes $\mathrm{e}^{-\frac{ws_1t}{1-t}}\left(\frac{-wt}{1-t}\right)^{s_2}$ in terms of $t$. The singularity of the second factor is compensated by the first factor, which exponentially goes to zero as $t\to1$. Hence a function with singularities of this type, together with all of its $t$-derivatives, will be regular in terms of $t$. Consequently, we can transform (particular) $C^k$ functions into $C^\infty$ functions.

In order to test whether this procedure improves the numerical accuracy, we reformulate the wave equation in terms of $t$. The result is
\begin{eqnarray}
  \frac{1}{w}\left[2-(1-\tau)A\right](1-t)^2\psi_{,tt}\nonumber\\
  -\left[\frac{1}{w}\left(2-(1-\tau)A\right)(2-w-2t)+2\left(1-(1-\tau)A\frac{1-2r}{1-r^2}\right)\right]\psi_{,t}\nonumber\\
  +2\rho\psi_{,t\rho}+\frac{w(1-\tau)}{(1-t)^2}A\,l(l+1)\psi
 +\frac{4rAw(1-\tau)}{(1+r)^2(1-t)^2}\psi=0,
\end{eqnarray}
with $A$ and $r$ as defined in \eref{eq:defA}, and where $\tau$ is considered to be a function of $t$. In the limit $t\to1$, this equation reduces to $\psi_{,t\rho}=0$, which we impose as condition at the upper boundary $t=1$ --- with exception of the point $t=1$, $\rho=0$, where we use the condition $\psi_{,t}=0$ instead. (Due to the structure of our new time coordinate, \emph{all} derivatives of $\psi$ that contain at least one $t$-derivative must vanish at $\mathscr{I}^+$, so it is clear that these conditions must hold.)

In the following, we solve the reformulated wave equation for solutions with different degrees of regularity, in order to test in which cases the new coordinates improve the numerical performance, and to find appropriate values for the parameter $w$.
\begin{figure}\centering
 \includegraphics[width=7.6cm]{expcoords1X.pdf}\quad
 \includegraphics[width=7.6cm]{expcoords2X.pdf}\\[1ex]
 \includegraphics[width=7.6cm]{expcoords3X.pdf}\quad
 \includegraphics[width=7.6cm]{expcoords4X.pdf}
 \caption{Comparison of the numerical accuracy that can be achieved with the new time coordinate $t$ for different choices of the parameter $w$, and the accuracy that is obtained with the old time coordinate $\tau$. The four examples (a)--(d) correspond to different initial data and  different values of the parameter $l$  as described in the main text.\label{fig:exp}}
\end{figure}
The convergence plots for four different numerical examples are shown in Fig.~\ref{fig:exp}. In the following we explain and discuss these examples.

In Fig.~\ref{fig:exp}a we consider the case $l=1$ and choose initial data \eref{eq:ab1} with $\alpha=0$, $\beta=1$. For this choice of parameters, the regularity conditions for $\psi_0,\dots,\psi_3$ in Table~\ref{tab} are all satisfied. Consequently, the first logarithmic term appears at sufficiently high order to allow for highly accurate numerical results in the old coordinates (with relative error below $10^{-12}$). When we solve the same initial value problem with the new coordinates, then we find that we lose about one order of magnitude in accuracy for the best value of $w$, and even more otherwise. Hence the new coordinates are not a good choice to resolve the ``relatively well-behaved'' function $\psi$ in this example.

The convergence plot for the same $l=1$ family of initial data as before, but now with parameter values $\alpha=5$, $\beta=1$, is shown in Fig.~\ref{fig:exp}b. Since $\alpha\neq0$ here, singularities already appear at a lower order than in the previous example. This reduces the accuracy that can be achieved with the old coordinates by about two orders of magnitude (with a relative error of approximately $10^{-10}$ for the largest considered resolution). For an appropriate choice of $w$, the new coordinates can achieve a better accuracy (with error around $10^{-11}$), and this is obtained for an even smaller number of collocation points. Hence the new coordinates provide a certain advantage  both in terms of efficiency and accuracy in this example.

Next, in Fig.~\ref{fig:exp}c, we consider the case $l=3$ with initial data \eref{eq:ab3}, and we choose $\alpha=5$, $\beta=1$. Using the old coordinates, the numerical error is $10^{-6}$ in the optimal case, which is still reasonably accurate, but far from the high accuracy that one usually expects from (pseudo) spectral methods. With the new coordinates, however, we can reduce the error to below $10^{-11}$ for appropriate values of $w$, i.e.\ we can improve the accuracy by five orders of magnitude.

Finally, for the example in Fig.~\ref{fig:exp}d, we choose $l=2$ and consider the initial data
\begin{equation}
 \tau=t=0:\quad \psi=\rho(\rho+1),\quad \psi_{,\tau}=0.
\end{equation}
These data satisfy the regularity condition for $\psi_0$, but not the one for $\psi_1$ in Table~\ref{tab}. This means that the mixed derivative $\psi_{,\tau\rho}$ will diverge along the cylinder in the limit $\tau\to1$. Similar singularities are expected at $\mathscr{I}^+$. With these properties, the solution is too singular to be recovered in terms of the original coordinates: the smallest relative error is only slightly below $10^{-3}$, which is not an acceptable level of accuracy to take this as a reliable numerical result (at least from the point of view of someone whose work goes beyond low-order finite difference methods). On the other hand, if we solve the same problem with the new coordinates, we can achieve an error below $10^{-12}$. Hence the new coordinates are essential to solve initial value problems that lead to very singular solutions, in which case the accuracy is immensely improved.

In summary, we observe that the original coordinates should be preferred whenever lower-order singularities are eliminated by appropriately restricting the initial data. However, when more singular solutions are considered, the new coordinates are extremely valuable in terms of the efficiency and accuracy that can be achieved. In particular, some solutions are too singular for a reasonable treatment with the old coordinates, but can be described very accurately with the new coordinates.

Finally, in all considered examples, we observed that the parameter $w$ in the new coordinates should be chosen somewhere near $w=3$ for the best results.

\section{Discussion\label{sec:discussion}}
As a toy model for physical equations like the conformal field equations and, in particular, in order to understand the behaviour of solutions near spacelike infinity, the conformally invariant wave equation on a Schwarzschild background was investigated. We did not impose any symmetry requirements on the solutions $\psi$ and hence extended the earlier discussion of spherically symmetric solutions \cite{FrauendienerHennig2017} to the general case. With a decomposition into spherical harmonics $Y_{lm}$, the $3+1$ dimensional problem can be reduced to $1+1$ dimensional equations, which turn out to depend on $l$ but not on $m$. These equations \JH{are} degenerate on $I$, the cylinder representation of spacelike infinity $i^0$, which allows \JH{us} to derive ordinary differential equations for $\psi$ and its spatial derivatives at $I$. \JH{An analysis} of these ODEs revealed that these functions generally suffer from logarithmic singularities at infinitely many orders. However, the solutions to the ODEs also allowed us to relate the singularities at time $\tau=1$ to the initial data at $\tau=0$. In particular, we can choose initial data subject to appropriate regularity conditions, such that the corresponding time-evolution of these data does not contain the first singular terms.

We also demonstrated that the fully pseudospectral time-evolution scheme can successfully be applied to the problem in question, and that highly accurate solutions can be obtained. An analysis of the numerical solutions showed that singularities are not restricted to the cylinder $I$, but that they generically also appear everywhere on future null infinity $\mathscr{I}^+$. The observed fall-off behaviour of certain Chebyshev coefficients is consistent with logarithmic singularities at various orders.

\JH{In addition}, we have studied how the order at which the first singularities appear influences the accuracy of the numerical results. If singularities are not present at very low orders, then highly accurate solutions close to machine accuracy are directly obtained. For ``more singular'' solutions, one can significantly improve the accuracy with an adapted new time coordinate. The corresponding coordinate transformation \JH{ensures} that the solutions are infinitely differentiable with respect to the new time coordinate, even though they have logarithmic singularities with respect to the old coordinate. After this modification, highly accurate results are again obtained.

As discussed, our considerations show that solutions to the conformally invariant wave equation generally have \JH{limited regularity}, both on the cylinder $I$ and at $\mathscr{I}^+$. The same is expected for solutions to more realistic equations. Nevertheless, \JH{as with the} present investigations, we predict that the corresponding initial value problems can be tackled with the fully pseudospectral method --- provided that one pays careful attention to the singularity structure and uses suitable formulations of the problems.

\section*{Acknowledgments}
 We would like to thank \JH{Matt Parry} for commenting on the manuscript. Part of this research was supported by the University of Otago through a Division of Sciences Strategic Seeding Grant.

\section*{References}

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Home » Uncategorized » Download Rope Hero (MOD, Unlimited Money) free on android
Download Rope Hero (MOD, Unlimited Money) free on android
Posted on: May 28, 2019 | By: harnamchema – Leave a Comment
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Basically, Rope Hero: Vice Town is a classic RPG game, not mixed with any extra elements. For example, a turn-based game is also labeled as an RPG or an RPG puzzle game. In our game, you have full control over the character, controlling all movement and personality.
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The controller of the game is quite familiar with the analog stick and the skill set on the right. Rope Hero is sure his main weapon will be a rope. The skill set is entertaining when the hero does not have the magical damage that is great. In contrast, he uses cables and adds a gun. But if you do not own a gun, then that is fine. Rope Hero has a remarkable physical strength thanks to the supernatural suit. This person can use his hand to punch the wall or kick a car very softly. He also can move fast between buildings by using ropes.
Also, when you do not fight, you must upgrade equipment and attributes for your character. First, increase the stats, such as damage, stamina, speed, … and then do not forget to equip heavy-duty guns. In addition to things to increase the power directly, the suit also has its own variations to make the player feel interesting.
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For the 21st installment in Tamarie Cooper's series of wildly popular summer shows, Tamarie is getting back to her roots. The first of her shows was one of the earliest instances of 'immersive theatre' in Houston, way back in 1996. Her second show was legendary for taking the audience on a field trip on an actual school bus to experience performances in the unlikeliest places. This year Tamarie's bringing back the bus and you're all invited!
Audiences will board the bus from Catastrophic's home base, MATCH, then travel around town and deep into Tamarie's own twisted psyche, dreams and lamentations. Tour a gallery of bad ideas! Take a whirl around Tamarie's mid-life crisis! Go to the bathroom with Tamarie! Original songs, a live band, tap dancing, games, tasty treats, and a motley crew of some of Houston's finest, funniest performers are all part of the 21st of Tamarie's original summer musicals. Make sure to bring your permission slips. This will be one amazing ride!
Formed in 2007 by Jason Nodler, Tamarie Cooper, The Catastrophic Theatre is Houston's premier creator and producer of new work for the theatre. The nationally acclaimed organization is dedicated to developing productions that foster a meaningful exchange between artists and audiences. A recipient of two prestigious MAP Fund commissioning grants, the theatre has gained an international reputation for its original pieces, which have attracted audiences from across the United States, Europe, and Australia. Artistic director Jason Nodler is the recipient of a NEA/MacDowell Colony fellowship, a Houston Press Mastermind award, and an unsolicited and unrestricted creative grant from Creative Capital. Named Houston's Best Theatre Company by the Houston Press in 2009, The Catastrophic Theatre offers Houston audiences a repertoire of challenging, innovative work that can't be seen anywhere else in the country.


text:
Inside the 'un-pc' social club off the beaten track
The Railway Club in Pennycomequick is a barrel of laughs and it's all thanks to Denise and her team
Miles O'LearyChief Reporter
Committed - Railway Club officials Colin Parker, Denis Hawking and Alan Day (Image: Miles O'Leary)
Down in Pennycomequick, there's a secret social club hidden round the back of a busy street - and it's got a blinding story to tell.
The Railway Club is a members-only den behind Stuart Road - which also happens to be the home and name of Plymouth's oldest clubhouse.
But it's the Railway that caught our fascination - simply because it's off the beaten track and undoubtedly a proper gem loved by the locals.
The entrance to the building - which from the outside would pass as a scout hut or industrial unit - is tucked away down in between the Speedy Depot.
I was met at the entrance door by a cheery bloke who so happened to be the treasurer, Alan Day.
He ushered me inside for a chat with colourful barmaid and steward Denise Hawking.
Character - Denise works five nights a week at The Railway Club (Image: Miles O'Leary)
Legend Denise - brimming with personality - has worked in the club and pub trade for 20 years.
What's made apparent very quickly is she's game for a laugh, doesn't take anything to heart - but would shoot you down in a flash at the bar if you were causing trouble or dropping F bombs.
The humbling stories behind Plymouth's most bombed locations
"It's the people, definitely the people that make the place what it is," Denise says, smiling.
"There's never any trouble in here. Everyone gets on with everybody.
"We all have a laugh and a joke; we're not a PC club."
Denise and her team love a laugh - but work hard to keep the customers happy (Image: Miles O'Leary)
Secretary Colin Parker chimes in: "We're leading the fight against political correctness.
"But if someone were to play up, or doesn't shut up, Denise would hit them!
Colin says much of the clientele are elderly and retired. But that doesn't make them stuffy or boring, not by a long shot.
He says many love a naughty giggle - that's just the habits of the older generation.
"I was in Majorca (on holiday) and one evening, we were doing bingo," Colin explains. "It was in a massive place, twice the size of this.
Proud - Colin and Alan in the main bar (Image: Miles O'Leary)
"A little guy from Liverpool came up with 69, meal for two, 69. The elderly ladies on the other table were screaming!"
Denise says one elderly local loves getting cheeky with a few of the female OAPs.
Plymouth nightclub could close because it can't pay £4,000 bill
But it's all good-natured, innocent fun and they enjoy the attention.
"When you get the older generation in, they are not precious or don't take offense, like some of the younger people do, but I don't," Denise said.
Watch a video below of the re-launched Seymour Arms pub in Plymouth
All jokes aside, the club prides itself on being a 'community centre' for the community.
There's loads of things going on during the week: dancing, bingo, dominos euchre - Colin also DJs.
He was setting up the sound system for a 93rd birthday party when we swung by.
Next week he's hosting a 21st and has ordered in some 'drum and bass' so the young ones have something current to bop their heads to.
The committee is also hoping to adjust their licence so they can host band nights for non-members in an effort to attract new blood.
The club's worth a visit simply to get a feel for its history.
It sprung to life in 1939, the year the Second World War broke out, and was a social bolt hole for British Rail workers.
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They kipped in the sheds outside - which also doubled up as kitchens - after their shifts and would come in the club for a pint.
Links to the railway gradually dwindled through the years and back in the 1990s the venue was seized back by members after the rail network was privatised.
Nowadays, there's only ONE Railway Club member left with links to British Rail, that being the club's president, ex-railway worker Bob Thomas.
But there's some 350 members still on the books enjoying good times.
Denise has spent years serving and entertaining people at the bar (Image: Miles O'Leary)
Devonport lass Denise, who grew up in St Budeaux, is one of the reasons why so many continue to pour through the doors.
Why? Because she's the heart and soul of the party and loves a good 'yap'.
"I love the camaraderie, the banter, the friendships," she said. "I love the whole ambiance.
"I used to work for a holiday camp - I was the bar and restaurant manager. New owners took it over, they told me I spent too much time talking to customers!
"I love all that, the yap. It's important in the pub trade."
Want Plymouth Live to write about your favourite pub or social club? Contact Miles on 01752 293 066 or email miles.oleary@reachplc.com
Pennycomequick
PubsInside the special place where Plymouth's bus drivers go to drinkLocally revered Pennycross Sports and Social Club is riding the wave of momentum after splashing the cash on modern touches
What's On56 Face of Plymouth male and female entries from 2007 - 2008
EntertainmentTop-secret Speakeasy event coming to Plymouth in unknown locationThe Secret 1920s Speakeasy will be held on Saturday, February 29 at an unknown location
Things to do PlymouthEverything we have to look forward to in Plymouth in 2020It's going to be an amazing year for Plymouth


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Last updated: 05:00 PM ET, Tue March 15 2016
GM and Lyft Drop Intriguing Express Drive Service On The Masses
Travel Technology	Gabe Zaldivar March 15, 2016
Photo courtesy of Thinkstock
Lyft and General Motors have unveiled their initial plans for what will become something like a rental-car, ride-sharing mix tape of a service.
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TechCrunch's Ingrid Lunden pulls back the curtain on what these two crazy kids have been doing since coming together for what was a $500 million GM investment at the time.
According to the report, the tandem are rolling out something called Express Drive, which will hit the Chicago market first, followed shortly thereafter in Boston, Washington D.C. and Baltimore.
From first glance, it seems that the burgeoning service is a way to answer the immediate needs for both GM and Lyft.
Lunden explains the service: "GM will provide all-in rental cars to Lyft drivers, who will pay between $99/week plus mileage and nothing at all, depending on how many Lyft rides they provide using the vehicles."
As noted, the first car to get the Express Drive treatment will be the Chevy Equinox.
So imagine scores of potential Lyft drivers coming to the realization that they can now use someone else's car for a nominal fee and get paid to do so.
For GM, the move makes tremendous sense as locals and tourists are whisked about town in a new model of a car with which they may just find themselves enamored.
Win, potentially, may just meet up with Win in this scenario.
Lyft co-founder John Zimmer explained to reporters, via Lunden, that there is a giant void of potential drivers that have the will but not the way when it comes to becoming a Lyft driver.
Lunden quotes the co-founder who expounded on the news: "We've now made car ownership optional on both sides of the market. Now you don't need to own a vehicle to make money on the platform, or to give rides to passengers."
The move comes in conjunction with GM's rollout of its Maven innovation, which TravelPulse's Donald Wood explained back in January: "GM's ride-sharing service will be similar to existing brands like Zipcar and Car2Go in that it will offer customers access to vehicles that can be rented quickly via a mobile app and then picked up from locations throughout a designated area."
Despite the advent of services of Uber and Lyft working their way into the mainstream, we are still very much in the initial throes of a new era wherein taxi services are on the way out and self-driving cars are on the foreseeable horizon.
GM's plan seems to be to provide a serviceable fleet that can meet the growing demand of travelers in major metropolitan areas who need immediate transport while maintaining a strong brand presence in this particular section of the industry.
Lyft's legion of dedicated drivers will become more resplendent as they roll out in their new Equinox cars.
For more Travel Technology News
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Experience More of Everything


Welcome to our website for all Made a hole with a tool Answers. This game is created by Fanatee for both iOS and Android devices.This game is perfect for iPhone, iPad, iPod Touch and Android devices and can be downloaded from both the iTunes App Store or Google Play Store. You cant find the solution? In our website you'll find the Answer.


text:
Harvey Stein (American, born 1941). The Hug: Closed Eyes and Smile, 1982.
Digital, inkjet archival print, 13 x 19 in. (33 x 48.3 cm).
Collection of the artist. © Harvey Stein, 2011
New York • Art & Culture
Relive the ups and downs of Coney Island with a look into the resort's definitive cultural history...
Rob Wilkes • 7 October, 2015
Frederick Brosen (American, born 1954). Fortune Teller, Jones Walk, Coney Island, 2008. Watercolor over graphite on paper, 17 7/8 x 11 ¼ in. (45.4 x 28.6 cm). Courtesy of Hirschl & Adler Modern, New York. Photo: Joshua Nefsky, courtesy of Hirschl & Adler Modern, New York; © 2013 Frederick Brosen/Artists Rights Society (ARS), New York
Coney Island's future is very much up in the air — although not in the way most New Yorkers would like. The historical resort, located on a peninsula at the southern tip of Brooklyn, once pulled huge crowds of holidaymakers keen to escape the oppressive summer heat of the inner city.
The amusement parks of Coney Island were enormously popular in the first half of the 20th Century, but the advent of cheap air travel led to ever-dwindling visitor numbers, and many of the old amusement parks now lie derelict.
Much of Coney Island is in the hands of developers with a history of undelivered regeneration promises, large parts of the real estate has been rezoned for housing use, and property damage caused by Hurricane Sandy in 2012 may well have been the final straw for Coney Island as a seaside entertainment destination.
Time for a bit of much-needed nostalgia! The Brooklyn Museum exhibition Coney Island: Visions of an American Dreamland, 1861-2008 begins at the beginning, looking at the resort's heyday through the eyes of a whole host of artists who fell in love with Coney Island both as a physical experience and as an escapist ideal. The show includes 42 photographs from the Brooklyn Museum collection, along with a wonderful collection of paintings, drawings, posters, film, salvaged objects and vintage souvenirs.
Put on your best boater and promenade to the exhibition beginning on 20 November, ending March 2016.
@brooklynmuseum
Strobridge Lithographing Company. The Barnum & Bailey Greatest Show on Earth /The Great Coney Island Water Carnival /Remarkable Head-Foremost Dives from Enormous Heights into Shallow Depths of Water, 1898. Color lithograph poster, 30 1/6 x 38 3/4 in. (76.6 x 98.4 cm). Cincinnati Art Museum; Gift of the Strobridge Lithographing Company, 1965.829
Steeplechase Funny Face, n.d. Painted metal, 23 3/8 x ¾ in. (59.4 x 1.9 cm). Collection of Ken Harck
Morris Engel (American, 1918–2005). Coney Island Embrace, New York City, 1938. Gelatin silver print, 10 9/16 x 11 ½ in. (26.8 x 29.2 cm). Orkin/Engel Film and Photo Archive. © Morris Engel
Edward J. Kelty (American, 1888–1967). Harlem Black Birds, Coney Island, 1930. Photograph, 12 x 20 in. (30.5 x 50.8 cm). Collection of Ken Harck. © Edward J. Kelty
Reginald Marsh (American, 1898–1954). Wooden Horses, 1936. Tempera on board, 24 x 40 in. (61 x 101.6 cm). Wadsworth Atheneum Museum of Art, Hartford, Connecticut; The Dorothy Clark Archibald and Thomas L. Archibald Fund, The Krieble Family Fund for American Art,The American Paintings Purchase Fund, and The Ella Gallup Sumner and Mary Catlin Sumner Collection Fund, 2013.1.1. Photo: © 2013 Estate of Reginald Marsh/Art Students League, New York/Artists Rights Society (ARS), New York
Samuel S. Carr (American, 1837–1908). Beach Scene, circa 1879. Oil on canvas, 12 x 20 in. (30.5 x 50.8 cm). Smith College Museum of Art, Northampton, Massachusetts; Bequest of Annie Swan Coburn (Mrs. Lewis Larned Coburn)
Red Grooms (American, born 1937). Weegee 1940, 1998–99. Acrylic on paper, 56 1/8 x 62 in. (142.6 x 157.5 cm). Private Collection. Photo: Courtesy of Marlborough Gallery, New York; © 2013 Red Grooms/Artists Rights Society (ARS), New York
Pip and Flip, 1932. Tempera on paper mounted on canvas, 48 1/4 x 48 1/4 in. Daniel J. Terra Collection, 1999.96.
Henry Koerner (America, born Austria, 1915–1991). The Barker's Booth, 1948–49. Oil on Masonite, 26 x 40 ½ in. (66 x 102.9 cm). Collection of Alice A. Grossman
Marie Roberts (American, born 1954). A Congress of Curious Peoples, 2005. Acrylic on unstretched canvas, 84 x 120 in. (213.4 x 304.8 cm). Collection of Liz and Marc Hartzman
Requiem for a Dream, production still, directed by Darren Aronofsky, 2000. Artisan Entertainment. Photo: Artisan/Photofest; © Artisan
Homer Page (American, 1918–1985). Coney Island, July 30, 1949. Gelatin silver print, 11 x 14 in. (27.9 x 35.6 cm). The Nelson-Atkins Museum of Art, Kansas City, Missouri; Gift of the Hall Family Foundation, 2008.47.6. ©Homer Page. Photo: John Lamberton
Little Fugitive, production still, 1953. Gelatin silver print, 8 x 10 in. (20.3 x 25.4 cm). A Morris Engel Production. Photo: Joseph Burstyn/Photofest; © Joseph Burstyn, Inc.; © Morris Engel
Photography, Exhibitions, Art, Life, History, Brooklyn, Culture, Coney Island
New book turns the knob and enters the private life of an unrivalled outsider art pioneer...
The genius of Gaudí informs the ornate feel of a design hotel that overlooks one of his most famous works...
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New York-based documentary photographer Peter van Agtmael looks at post-9/11 through a poignant lens in astonishing body of work...
In his poignant and inspired canon of work, New York-based documentary photographer, Peter van Agtmael, has cast a critical eye on the United States and its post-9/11 wars since 2006; receiving a Guggenheim Fellowship, several Pulitzer Center Grants...
Wooden doors to refrigerators and football helmets, few discarded objects would escape Basquiat at his most viscerally creative...
On view until 11 June, Jean-Michel Basquiat: Art and Objecthood provides an in-depth look into the artist's often overlooked sculptural practice. Curated by Dr. Dieter Buchhart, the exhibition is the first to be dedicated to the role of found...
From Paris, 1910, to 1950s São Paulo and beyond, Oscar Oiwa gives us an insight into his creative mind with a time travelling exhibition...
Brazilian-American visual artist, Oscar Oiwa, uses time travel, role-playing, and sheer artistic license in If I were living in, his latest exhibition currently on show at multipurpose hub for New York City-based Japanese creators, NowHere. A...


text:
set -o errexit -o pipefail

cd /

### VMWare Guest Utils installation ###

# Install guest utils
pacman -S --noconfirm open-vm-tools

# Enable the service
systemctl enable vmtoolsd.service

# Enable some VMWare kernel modules
mkdir -p /etc/modules-load.d
cat <<LIST > /etc/modules-load.d/vmware.conf
vmw_balloon
vmxnet3
LIST

# Prevent guest utilities from being upgraded (they might depend on newer kernel version and cause
# unresolvable conflict)
patch -p 0 -i /tmp/vmware/pacman.conf.diff

### Fixes for known errors ###

cat <<LIST >> /etc/modprobe.d/blacklist.conf
# Disable intel_rapl to stop errors about no valid rapl domains
blacklist intel_rapl

# Disable i2c_piix4 kernel module to stop complaining about uninitialised SMBus address
blacklist i2c_piix4
LIST

### Finalise... ###

# Regenerate ramdisk
mkinitcpio -p linux

redpajama_set_name: RedPajamaGithub


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text:
Jorge de la Vega Domínguez är en ort i Mexiko.   Den ligger i kommunen El Porvenir och delstaten Chiapas, i den sydöstra delen av landet,  km sydost om huvudstaden Mexico City. Jorge de la Vega Domínguez ligger  meter över havet och antalet invånare är .

Terrängen runt Jorge de la Vega Domínguez är bergig norrut, men söderut är den kuperad. Den högsta punkten i närheten är El Porvenir,  meter över havet,  km norr om Jorge de la Vega Domínguez. Runt Jorge de la Vega Domínguez är det ganska tätbefolkat, med  invånare per kvadratkilometer. Närmaste större samhälle är Motozintla de Mendoza,  km sydost om Jorge de la Vega Domínguez. Omgivningarna runt Jorge de la Vega Domínguez är en mosaik av jordbruksmark och naturlig växtlighet.

Årsmedeltemperaturen i trakten är  °C. Den varmaste månaden är april, då medeltemperaturen är  °C, och den kallaste är september, med  °C. Genomsnittlig årsnederbörd är  millimeter. Den regnigaste månaden är juni, med i genomsnitt  mm nederbörd, och den torraste är januari, med  mm nederbörd.

Kommentarer

Källor

Orter i Chiapas

redpajama_set_name: RedPajamaWikipedia


text:
February 5, 2009 Asia, Floods & Storms, News Headlines
Thousands stranded by intense Brunei floods
BANDAR SERI BEGAWAN, BRUNEI Feb 5, 2009 – Thousands of people in rural Brunei have been stranded by flood waters up to two metres deep, a newspaper reported on Thursday.
Heavy rains on Tuesday night rendered roads in Tutong district impassable, forcing some residents to use boats for transport, the Borneo Bulletin reported.
"Many couldn't get out of their villages because the main road is submerged under water while many were unable to return home," the Bulletin quoted Omar bin Abdul Azif, a village chief, as saying.
He added that students could not get to their classes.
Last month two people died after heavy rains caused landslides and flooding that were among the sultanate's worst ever, the Bulletin said.-AFP
Brunei Floods, Climate Change, Global Warming, Landslides, Stranded, Thousands, Tutong
Powered by WordPress | you could check here | Thanks to Learn More about Virgin Mobile, additional reading and he has a good point


I finished up the second Vanguard model, and I think I did a better job than the first. I had some issues with the varnishing and with the photography, but the model looks pretty good in person.
I did a much more thorough job highlighting the model overall, and I'm quite pleased with how it all came out. I also tried to make the base clearer, with better gradients. The biggest improvement is the face, which I reduced to a blobby mass on the first model. I finally learned just what people are talking about with multiple (really thin) thin layers of paint, so as not to ruin detail. I also toned down the front arc marking a lot from the prior model. It's probably a little too subdued now, but I'll fix that by having the highlighting go all the way up to white in the future, as it does for the hexes on the base.
Games and Stuff is running a Warmachine Journeyman league, and I will be building my Blindwater Congregation Gatorman army. I already own a number of the models I will need, although a few of them are already painted.
Deploy 2 3″ shallow waters.
Of those models, Calaban and the Ironback Spitter are already painted. I'll be working on the Wrastler starting this week.
I'm also going to start roughing out plans for the shallow water templates, as I've been doing recessed swamp bases, with plans to do use Envirotex Lite for water effects.
You are currently browsing the Home of the Ghostlords archives for February, 2012.


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THE FALL + WINTER '18-'19 COLLECTION IS HERE!
NEW PRODUCTS ARE LIVE + WAYS TO WEAR THEM!
THE SPRING + SUMMER '17 COLLECTION IS HERE!
THE MANY WAYS TO WEAR A SUGARSKY HEADBAND!
A NEW LOOK + KEEP US POSTED!
THE WINTER COLLECTION IS HERE!


The winter of 2011-12 is fast approaching, and on November 11, we made our first foray to a power-plant reservoir to pursue black and temperate bass. As the last 37 days of the autumn of 2011 winds down, we will be making more trips to fish the waters surrounding the power plants.
Eventually when our lakes become ice-covered for days on end, they will be the only nearby waterways where we can launch a boat. One of these lakes has a substantial population of medium-sized smallmouth bass. The other contains a hefty population of lunker-sized largemouth bass. Both are graced with wipers and white bass.
From about 1980 to 2002, we were convinced that versatility was the best piscatorial tactic for recreational bass anglers to employ. Therefore, we always tested a variety of power tactics, intertwined with occasional finesse presentations, on every winter outing. We were also part of the school that liked to wield big soft-plastic baits affixed to big jigs. In short, we were using the methods popularized by professional tournament anglers and the fishing media rather than the ones that were suitable for the conditions and waterways that we fished as recreational anglers. It is similar to playing backyard baseball and football and acting as if we are Albert Pujols and Peyton Manning.
It's often assumed that professional anglers catch prodigious numbers of bass. But if you were the proverbial mouse in the back of a talented tournament angler's boat, watching him as he practices for and competes in a tournament, you would soon realize how few bass he normally catch during several long, hard days afloat. The same observation can be made about the number of bass that are caught during the filming of a television show. In fact, one of the nation's best professional bass tournament anglers who also works for a popular TV show recently said that creating a 10-minute segment for a TV show is often more trying than fishing a tournament; the reason is that the theme of the show is usually focused on how he uses one of his most successful tournament or power-fishing tactics, and there are spells when the bass aren't readily allured by such a tactic, but across many hours of hard fishing, a 10-minute feature usually can be created.
Our perspectives about the best way for recreational anglers to catch black bass began to change shortly after the turn of the millennium. By 2003 we began to return to the finesse methods that the late Chuck Woods of Kansas City pioneered in the 1950s and 1960s. Woods developed a system that allures not only scores of bass, but some lunkers to boot, in the flatland waterways of northeastern Kansas and northwestern Missouri. By the way, Woods' biggest bass weighed 10 pounds, 3 ounces.
Nowadays Woods system is called Midwest Finesse, and since several of Woods' disciples started the Finesse News Network a few years ago, it has been determined that Midwest Finesse tactics work surprisingly well at various and asunder waterways across the nation.
Many anglers, however, are uncomfortable utilizing finesse tackle and presentations, but wintertime outings at power-plant reservoirs are an excellent time and place for novice finesse anglers to get acquainted or comfortable with the manifold virtues of finesse fishing for bass. Once these novices get hang of it, they will soon see that a pair of knowledgeable finesse anglers can follow a virtual phalanx of power anglers along a shoreline or around a series of points and offshore humps, and they will inveigle an amazing number of largemouth bass that the power anglers fail to allure.
During some winter outings at our power-plant reservoirs, there are many boatloads of anglers plying the same shorelines, points and offshore lairs. At times it is a virtual traffic jam along some locales. Despite that chaos and angling pressure, our finesse tactics normally allure a surprising number of bass.
Besides catching wintertime bass that power anglers can't catch, another virtue of Woods' finesse tactics is that it is relatively inexpensive, and in these trying economic times, that's a sterling attribute. For instance, inexpensive medium-action spinning outfits spooled with either eight or 10-pound-test monofilament line work as well as state-of-the-art spinning outfits spooled with top-of-the-line and high-dollar eight- or 10-pound-test braided line and fluorocarbon leader.
In regard to braided line, several Midwest finesse anglers have been using Zebco's rather inexpensive 10-pound-test Omniflex braid for the past year, and these anglers have been pleasantly surprised with its effectiveness.
What's more, day in and day out, we use nine styles of soft-plastic lures that are affixed to a jig, and they are relatively inexpensive and astonishingly durable.
Last winter at the power-plants, the three most fruitful lures were made by Z-Man Fishing Products; they were the four-inch Finesse ShadZ, three-inch Rain MinnowZ, and a customized 2 1/2-inch ZinkerZ. The ZinkerZ is normally five-inches long; it's customized by merely cutting it in half. These three soft-plastic baits were affixed to either a 1/32-ounce or 1/16-ounce Original Mushroom Jig Head made by Gopher Tackle Manufacturing Company.
And on our initial cold-water outing this autumn, a purple-haze 2 1/2-inch ZinkerZ paid the best dividends. So we anticipate the Finesse ShadZ , Rain MinnowZ and ZinkerZ will be fruitful throughout the winter of 2011-12.
In winters past, we have noticed that most anglers ply the lairs that are situated within the plum of warm water that is jettisoned out of a power plant. We have found, however, that many of the cold-water lairs that lie outside of the water-water plum produce a surprising number of largemouth bass as along as the surface temperature is above 40 degrees or a tad warmer.
As our cold-water season unfolds, we will delve into the archives and post a piece about how to use finesse tactics to catch smallmouth bass that abide in a power-plant reservoir. We will also post a story about how Brian Waldman of Coatesville, Indiana, uses a tiny hair jig and plastic chunk to entice cold-water bass. In addition, we will dissect the various retrieves that we employ with of our soft-plastic-and-jig combos.


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21 December 2022 by AHB Library Published in: • The Plural Eldership Issue, George Ella, Doctrinal Matters No comments yet
An Extract From George Ella's Book, "John Gill And The Cause Of God And Truth", Pages 54,55. A Printed Copy Of This Book May Be Obtained From Go Publications (£13/$16).
The records of Gill's ordination service have provided church historians with a complete picture of how the Particular Baptists were organized at the beginning of the eighteenth century. It is clear that they only elected one elder per church, who was also their pastor, who then presided over a number of deacons.[1] Though many pastors were present at Gill's ordination ceremony, they were referred to by Crosby in his records as 'elders'. This might indicate that the Baptists had some idea of a church universal with elders belonging to it which were not members of the local church but fellow members of the body of Christ. In their confessions of faith, however, they tended to define the local church as if it excluded any membership of a wider church concept. This wider church concept, however, in many Baptist churches, had begun to take the form of fraternals, associations, societies and unions. They, in turn, tended to isolate the normal church members from fellowship with the other churches as only representatives called 'messengers' took an active part in inter-church dealings. Gradually these associations and unions, with their representative membership only, elected non-church-based hierarchies with no doctrinal creeds who saw themselves as ruling bodies over local churches.[2] Gill was to do much in his pastorate to curb this downgrading of a true church conception but he was very much a lone voice and his successor John Rippon went back to the old para-church policies. By the time Charles Haddon Spurgeon became pastor of Gill's church after Rippon, over a hundred years later, the situation had become so bad that a doctrineless Baptist Union dominated the majority of local churches, ruling them from outside. Churches founded abroad by British missionaries were ruled by a committee of men in the Baptist Missionary Society who opened their membership to anyone who would pay 10/6 , irrespective of their beliefs. Declaring that he was taking up the mantle cast down at Gill's death, Spurgeon took up the postponed fight again for the rights of the local church over external organizations, patronages and societies. Spurgeon had a measure of success in his reforming plans for the Baptist Missionary Society but he failed sadly in his opposition to the Baptist Union who continued to put organizational unity sealed by the rite of baptism before a common faith.
[1] Ivimey gives the exception of a Bristol church that traditionally elected a co-pastor. See vol. iv, pp. 283-289.
[2] There were exceptions regarding statements of faith. The London Particular Baptist Board demanded that its members be Calvinists and the Western Association required their members to sign the 1689 Confession from around 1733 onwards. In 1782, for instance, the Kent and Sussex Association headed their Association Letter with a brief statement of faith.
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