text:
\section{Introduction}

This paper is concerned with high-order robust numerical schemes for hyperbolic conservation laws
\begin{equation}\label{HCL}
	\begin{cases}
	u_t + \nabla \cdot {\bm f} (u) =0,  \quad  & ({\bm x},t) \in \mathbb R^d \times \mathbb R^{+},
	\\
	u({\bm x},0) = u_0({\bm x}), \quad & {\bm x} \in \mathbb R^d,
\end{cases}
\end{equation}
where ${\bm x}$ denotes the spatial coordinate variable(s) in $d$-dimensional space, $t$ denotes the time,
the conservative variable(s) $u$ takes values in $\mathbb R^m$, and the flux ${\bm f}=(f_1,\dots,f_d)$ takes values in
$(\mathbb R^m)^d$. Our discussions in this paper can also be applicable to
other related hyperbolic or convection-dominated equations.

Solutions to the hyperbolic equations \eqref{HCL} often satisfy certain bounds, which constitute a convex invariant region $G \subset \mathbb R^m$.
When numerically solving such hyperbolic equations, it is highly desirable or even essential to
preserve the intrinsic bounds, namely, to preserve the numerical solutions in the region $G$.
In fact, if the numerical solutions go outside the bounds, for example, negative density or pressure is produced
when solving the Euler equations, the discrete problem would become ill-posed due to the loss of hyperbolicity of the system,
and may lead to the instability or breakdown of the numerical computation.

As well known, first-order accurate monotone schemes,
 such as the Godunov scheme, the Lax--Friedrichs scheme, and
the Engquist--Osher scheme, are bound-preserving (BP) for scalar conservation laws and many hyperbolic systems.
However, seeking high-order accurate BP schemes is rather nontrivial.
In the pioneering work of \cite{zhang2010,zhang2010b}, Zhang and Shu proposed a general framework of designing high-order BP discontinuous Galerkin (DG) and  finite volume (FV) schemes for hyperbolic conservation laws on rectangular meshes,  later generalized to triangular meshes in \cite{zhang2012maximum}.
Over the past decade, the Zhang--Shu framework has attracted extensive attention and been applied to various hyperbolic systems (e.g., \cite{xing2010positivity,zhang2011,WangZhangShuNing2012,zhang2012minimum,QinShuYang2016,Wu2017,jiang2018invariant,du2019high,wu2021minimum})
and convection-dominated equations
 (e.g., \cite{zhang2012maximum-b,zhang2013maximum,ZHANG2017301,sun2018discontinuous,du2019maximum}).
Recently, motivated by a series of BP works \cite{WuTangM3AS,Wu2017a,WuShu2018,WuShu2020NumMath}
for magnetohydrodynamics,
the geometric quasilinearization (GQL) framework was proposed in  \cite{Wu2021GQL} for studying BP problems involving nonlinear constraints.
For more developments on high-order BP schemes, we refer the reader to the review papers \cite{zhang2011b,XuZhang2017,Shu2020AC} and some other BP techniques \cite{Xu2014,xiong2016parametrized,WuTang2015,guermond2017invariant}.

An essential ingredient in the Zhang--Shu framework \cite{zhang2010,zhang2010b} is to decompose the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points.
Based on such a convex decomposition, one can reformulate a one-dimensional (1D) or multidimensional high-order FV or DG scheme into a convex combination of formally 1D first-order schemes. This reformulation then leads to a sufficient condition for the BP property of the updated cell averages, combined with a
simple local scaling limiter that can enforce the sufficient condition without losing the high-order accuracy \cite{zhang2010,zhang2010b}.

To illustrate the role of convex decomposition in Zhang--Shu's BP framework, let us consider
a $(k+1)$th-order FV or DG scheme for 1D conservation laws with reconstructed or DG polynomials
of degree $k$, denoted by $p_i(x)$, on cell $\Omega_{i}=[x_{i-\frac12}, x_{i+\frac12}]$.
With forward
Euler time discretization, the evolution equation of cell averages
can be written as
\begin{equation}\label{ZS-1D-cell-ave-evo}

	\bar {u}_i^{n+1} = \bar u_{i}^n - \lambda
	\left(  \hat f( u_{i+\frac12}^{-}, u_{i+\frac12}^{+}  ) - \hat f( u_{i-\frac12}^{-}, u_{i-\frac12}^{+}  )  \right),
\end{equation}
where $\bar u_{i}^n$ is the cell average of $p_i(x)$ on $\Omega_{i}$ at time level $n$,
$\lambda = \Delta t/\Delta x$ is the ratio of the temporal and spatial step-sizes, $u_{i-\frac12}^{+} = p_i( x_{i-\frac12} )$ and
$u_{i+\frac12}^{-} = p_i( x_{i+\frac12} ) $ for all $i$.
Here, $\hat f(\cdot,\cdot)$ is a BP numerical flux with which the first-order scheme is BP under  a suitable CFL condition
$
	a  \lambda \le c_0,
$
where $a$ denotes the maximum characteristic speed, and $c_0$ is the maximum allowable CFL number for the corresponding first-order scheme.
Note that the $L$-point Gauss--Lobatto quadrature with $L=\lceil \frac{k+3}{2}\rceil$ is exact for polynomials of degree up to $k$. This implies the following convex decomposition \cite{zhang2010}:
\begin{equation}\label{eq:GL1D}
	\bar u_i^n = \frac{1}{\Delta x} \int_{ x_{i-\frac12} }^{ x_{i+\frac12} } p_i(x) {\rm d} x
	= {\omega}_1^{\tt GL}  u_{i-\frac12}^{+}  +  {\omega}_L^{\tt GL}  u_{i+\frac12}^{-} +
	\sum_{s = 2}^{L-1}  {\omega}_\ell^{\tt GL} p_i(  x_{i,s}^{\tt GL} ),
\end{equation}
where  $\{ {\omega}_s^{\tt GL} \}$ are the Gauss--Lobatto weights with ${\omega}_1^{\tt GL}={\omega}_L^{\tt GL}=1/(L(L-1))$, and $\{  x_{i,s}^{\tt GL} \}$ are the quadrature nodes on $\Omega_{i}$ with $x_{i,1}^{\tt GL} = x_{i-\frac12}$ and $x_{i,L}^{\tt GL} = x_{i+\frac12}$.
Based on the decomposition \eqref{eq:GL1D}, Zhang and Shu \cite{zhang2010,zhang2010b}
rewrote the scheme \eqref{ZS-1D-cell-ave-evo}
 as
\begin{equation}\label{eq:1Dcd}
	\bar u_i^{n+1} =  {\omega}_1^{\tt GL}   \Pi_1 +   {\omega}_L^{\tt GL} \Pi_L  + \sum_{s = 2}^{L-1}  {\omega}_s^{\tt GL} p_i(  x_{i,s}^{\tt GL} ),
\end{equation}
where
\begin{equation*}
	\Pi_1 :=  u_{i-\frac12}^+ - \frac{ \lambda  }{ {\omega}_1^{\tt GL}  } \left(
	\hat f ( u_{i-\frac12}^+, u_{ i+\frac12  }^- )
	- \hat  f( u_{i-\frac12}^{-}, u_{i-\frac12}^{+}  )
	\right),
	 \qquad
	\Pi_L :=  u_{i+\frac12}^- - \frac{ \lambda  }{ {\omega}_L^{\tt GL}  } \left(
\hat f( u_{i+\frac12}^{-}, u_{i+\frac12}^{+}  )
- \hat f ( u_{i-\frac12}^+, u_{ i+\frac12  }^- )
\right)
\end{equation*}
are of the same form as the three-point first-order scheme with a scaled time step-size.
Thanks to the convex decomposition \eqref{eq:1Dcd} and the BP property of the first-order scheme,
 if
 we use a BP limiter \cite{zhang2010} to enforce
 \begin{equation}\label{eq:1DPPcond}
 	p_{i}(  x_{i,s}^{\tt GL} ) \in G  \qquad \forall i, s,
 \end{equation}
 then by the convexity of $G$, the high-order scheme \eqref{ZS-1D-cell-ave-evo} preserves $\bar u_j^{n+1} \in G$ under the CFL condition
 \begin{equation}\label{1Dhst_CFL}
 	a  \lambda \le c_0 \omega_1^{\tt GL}.
 \end{equation}

As we have seen, the convex decomposition \eqref{eq:GL1D} plays a critical role in
constructing high-order BP schemes. It determines not only the theoretical BP CFL condition \eqref{1Dhst_CFL} of the resulting scheme, but also the points   \eqref{eq:1DPPcond} to perform the BP limiter.
In fact, one may choose a different convex decomposition in the above analysis.
In the 1D case,
any type of quadrature rule, with weights all positive and nodes including the two endpoints,
would give a feasible convex decomposition.
However, different decomposition would affect the theoretical BP CFL condition and thus the computational costs. It is natural to ask what decomposition is optimal in the sense of
achieving the mildest BP CFL condition.
Zhang and Shu mentioned in \cite[Remark 2.7]{zhang2010} that they checked, for $k=2,3$, that their
1D decomposition \eqref{eq:GL1D} is optimal.
For $k\ge 4$, the optimality of the 1D decomposition \eqref{eq:GL1D} has not been proved yet.

This paper aims to make the first attempt at questing the optimal convex decomposition in the {\em multidimensional} cases. In the multiple dimensions, Zhang and Shu \cite{zhang2010,zhang2010b} proposed the classic
convex decomposition based on the tensor product of the Gauss--Lobatto quadrature and the Gauss quadrature rules; see \eqref{eq:411}.
As the 1D case, their decomposition has been an important foundation for constructing high-order BP multidimensional schemes.
Over the past decade, the classic Zhang--Shu decomposition
has been widely adopted in designing many high-order BP schemes for various hyperbolic or convection-dominated equations. It is natural to ask the following question:
\\[0.01mm]
\begin{center}
	\parbox{0.888\textwidth}{\em {
			Is the classic convex decomposition optimal in multiple dimensions?}\\[0.01mm]}
\end{center}
In this work, we find, in the {\em multidimensional} cases, that the classic decomposition is
generally not optimal for the $\mathbb P^k$ spaces, {\em i.e.}~the multivariate polynomial spaces of total degree up to $k$. {\em Seeking the optimal convex decomposition in the multidimensional cases is highly complicated and challenging.}
In this paper, we restrict our attention to two commonly used spaces ($\mathbb P^2$ and $\mathbb P^3$), which are typically used in the third-order and fourth-order DG schemes, on the Cartesian meshes. For these polynomial spaces, {\em we discover a novel alternative decomposition,
which is rigorously proved to be optimal, namely, to attain the mildest BP CFL condition, yet requires much fewer nodes.}
Based on our novel optimal convex decomposition, we can establish more efficient high-order BP DG schemes in the Zhang--Shu framework, as it allows a notably larger BP time step-size than the classic one.
The discovery of our optimal convex decomposition is highly nontrivial and may have a broad impact, as it would
lead to an overall improvement of third-order and fourth-order BP schemes for
a large class of hyperbolic or convection-dominated equations at the cost of only a slight and local modification to the implementation code.
We will present several numerical examples to further validate the remarkable advantages of using our optimal decomposition over the classic one in terms of efficiency.
It is worth mentioning that seeking the optimal convex decomposition for general $\mathbb P^k$ spaces ($k\ge 4$) in the multidimensional cases (on the Cartesian meshes or unstructured meshes) seems challenging and is still open.
We hope the present paper could be helpful for motivating further discussions on this interesting problem
in the future.

\section{General convex decomposition for 2D high-order BP schemes}\label{sec:2}

This section discusses the general feasible convex decomposition for constructing 2D high-order BP DG schemes within
the Zhang--Shu framework.

Let $\mathbb P^k$ denote the space of multivariate polynomials of total degree up to $k$.
We consider the $(k+1)$th-order $\mathbb P^k$-based DG scheme with the forward Euler time discretization for solving
 the 2D hyperbolic conservation laws
\begin{equation}\label{2DCL}
u_t+f_1(u)_x+f_2(u)_y = 0, \quad (x,y,t) \in \mathbb{R} \times \mathbb{R} \times \mathbb R^{+}.
\end{equation}
All our discussions in this paper are also valid for high-order strong-stability-preserving (SSP) time discretizations \cite{GottliebKetchesonShu2011}, as they are convex combinations of forward Euler step.
Following the Zhang--Shu framework \cite{zhang2010,zhang2010b}, in order to design a BP DG scheme, we only need to ensure the cell averages within the region $G$. As long as the BP property of the updated cell averages is guaranteed,
one may employ a simple BP limiter to enforce the pointwise bounds
of the piecewise DG polynomial solutions without affecting the high-order accuracy \cite{zhang2010,zhang2010b}.

On a rectangular cell $\Omega_{ij}:=[x_{i-\frac12},x_{i+\frac12}]\times[y_{j-\frac12},y_{j+\frac12}]$, the evolution equation of cell averages for the $(k+1)$th-order DG scheme reads
\begin{equation}\label{eq:171}
	\bar {u}_{ij}^{n+1} =
	\bar u_{ij}^n -
	\frac{\Delta t}{\Delta x} \sum_{q=1}^Q \omega_q^{\tt G}
	\left[  \hat f_1 \big( u_{i+\frac12,q}^{-}, u_{i+\frac12,q}^{+}  \big) - \hat f_1 \big( u_{i-\frac12,q}^{-}, u_{i-\frac12,q}^{+}  \big)  \right] -
	\frac{\Delta t}{\Delta y} \sum_{q=1}^Q \omega_q^{\tt G}
	\left[  \hat f_2 \big( u_{q,j+\frac12}^{-}, u_{q,j+\frac12}^{+}  \big) - \hat f_2 \big( u_{q,j-\frac12}^{-}, u_{q,j-\frac12}^{+}  \big)  \right],
\end{equation}
where
\begin{equation*}
	u_{i-\frac12,q}^{+} = p_{ij} \big( x_{i-\frac12}, y_{j,q}^{\tt G} \big), \quad
u_{i+\frac12,q}^{-} = p_{ij} \big( x_{i+\frac12}, y_{j,q}^{\tt G} \big), \quad
u_{q,j-\frac12}^{+} = p_{ij} \big( x_{i,q}^{\tt G}, y_{j-\frac12} \big), \quad
u_{q,j+\frac12}^{-} = p_{ij} \big( x_{i,q}^{\tt G}, y_{j+\frac12} \big)
\end{equation*}
with $p_{ij}(x,y) \in \mathbb{P}^k$ denoting the DG solution polynomial on $\Omega_{ij}$ at time level $n$ satisfying
\begin{equation*}
	\bar {u}_{ij}^{n} = \frac{1}{\Delta x \Delta y} \int_{ x_{i-\frac12} }^{ x_{i+\frac12} } \int_{ y_{j-\frac12} }^{ y_{j+\frac12} } p_{ij}(x,y) \, {\rm d} y {\rm d} x,
\end{equation*}
and $\{x_{i,q}^{\tt G}\}_{q=1}^Q$ and $\{y_{j,q}^{\tt G}\}_{q=1}^Q$ respectively denote
the $Q$-point Gauss quadrature nodes in the intervals
$[x_{i-\frac12},x_{i+\frac12}]$ and $[y_{j-\frac12},y_{j+\frac12}]$, with
the corresponding quadrature weights $\{\omega_{q}^{\tt G}\}$ satisfying
$\sum_{q=1}^Q \omega_{q}^{\tt G} =1$. For the $\mathbb P^k$-based DG scheme, $Q$ is typically taken as $k+1$, such that the quadrature has sufficiently high-order accuracy.

In \eqref{eq:171}, we take the numerical fluxes $\hat{f}_1$ and $\hat{f}_2$ as the BP numerical fluxes with which the corresponding 1D three-point first-order schemes are BP, {\rm i.e.}, for any $u_1,u_2,u_3 \in G$ it holds that
\begin{equation}\label{1DBP}
 u_2 - \frac{ \Delta t }{ \Delta x} \left( \hat{f}_1(u_2,u_3) - \hat{f}_1(u_1,u_2) \right) \in G, \qquad
 u_2 - \frac{\Delta t }{ \Delta y} \left( \hat{f}_2(u_2,u_3) - \hat{f}_2(u_1,u_2) \right) \in G \qquad
\end{equation}
under a suitable CFL condition $\max\{a_1 \Delta t / \Delta x,a_2 \Delta t/\Delta y\} \le c_0$, where $a_1$ and $a_2$ denote the maximum characteristic speeds in $x$- and $y$-directions, and $c_0$ is the maximum allowable CFL number for the 1D first-order schemes. For example, typically $c_0=1$ for the Lax--Friedrichs flux \cite{zhang2010b,jiang2018invariant},  and $c_0=\frac12$ for the HLL and HLLC fluxes \cite{jiang2018invariant}.

\subsection{Feasible convex decomposition in 2D}

Similar to the 1D case (\ref{eq:GL1D}), the BP analysis and design of a 2D scheme  (\ref{eq:171}) also require certain 2D quadrature rule to decompose the cell average $\bar u_{ij}^n$ into a convex combination of the values of $p_{ij}$ at some points.
A qualified 2D quadrature, which we call {\em feasible convex decomposition}, should satisfy three requirements, as defined below.

\begin{definition}[Feasible convex decomposition in 2D]\label{def:2D_FCAD}
	A 2D convex decomposition
	\begin{equation}\label{2Ddecomp}
			\bar {u}_{ij}^{n} =
		 \sum_{q=1}^Q \omega_{q}^{\tt G}  \left[
		 \omega_1^- u_{i-\frac{1}{2}, q}^{+} +
		 \omega_1^+ u_{i+\frac{1}{2}, q}^{-} +
		 \omega_2^- u_{q, j-\frac{1}{2}}^{+} +
		 \omega_2^+ u_{q, j+\frac{1}{2}}^{-}
		 \right]
		 + \sum_{s=1}^S \omega_s p_{ij} ( x_{ij}^{(s)}, y_{ij}^{(s)} )
	\end{equation}
	is said to be {\em feasible} for the polynomial space $\mathbb P^k$, if it simultaneously satisfies
	the following three conditions:
	\begin{enumerate}[label=(\roman*)]
		\item  the convex decomposition holds exactly for all $p \in \mathbb P^k$;
		\item  the weights $\{{\omega}_1^\pm, {\omega}_2^\pm, {\omega}_s\}$ are all positive (their summation equals one);
		\item  the internal node set $\mathbb S_{ij} := \{ ( x_{ij}^{(s)}, y_{ij}^{(s)}  ) \}_{s=1}^S \subset \Omega_{ij}$.
	\end{enumerate}
\end{definition}

Based on the tensor product of the $L$-point Gauss quadrature (with $L=\lceil \frac{k+3}{2}\rceil$) and the $Q$-point Gauss--Lobatto quadrature, the cell average $\bar {u}_{ij}^{n}$ can be decomposed into a convex combination of point values of $p_{ij}$ as follows:
\begin{align}\nonumber
\bar {u}_{ij}^{n}
& = \frac{1}{\Delta x \Delta y} \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} p_{i j}(x, y) \mathrm{d} x \mathrm{d} y
=
\sum_{s=1}^{L} \omega_{s}^{\tt {GL}}\left(\frac{1}{\Delta y} \int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} p_{i j}\left(x_{i, s}^{\tt {GL}}, y\right) \mathrm{d} y\right) =
\sum_{s=1}^{L} \sum_{q=1}^{Q} \omega_{s}^{\tt {GL}} \omega_{q}^{\tt {G}}p_{i j}\left(x_{i, s}^{\tt {GL}}, y_{j, q}^{\tt {G}}\right)
 \\ \label{eq:GGL_x}
&  =
\sum_{q=1}^Q \omega_{q}^{\tt G} \omega_1^{\tt GL} \left[
		 u_{i-\frac{1}{2}, q}^{+} + u_{i+\frac{1}{2}, q}^{-}
\right]
 +
\sum_{s=2}^{L-1} \sum_{q=1}^{Q}  \omega_s^{\tt GL}  \omega_q^{\tt G}
p_{i j}\left(x_{i,s}^{\tt GL}, y_{j,q}^{\tt G} \right)
.
\end{align}
Similarly,
by applying the quadrature rules in a different order, one obtains
\begin{equation}\label{eq:GGL_y}
\bar {u}_{ij}^{n} =
\sum_{q=1}^Q \omega_{q}^{\tt G} \omega_1^{\tt GL} \left[
		 u_{q, j-\frac{1}{2}}^{+} +
		 u_{q, j+\frac{1}{2}}^{-}
\right]
 +
\sum_{s=2}^{L-1} \sum_{q=1}^{Q}  \omega_s^{\tt GL}  \omega_q^{\tt G}
p_{i j}\left(x_{i,q}^{\tt G},y_{j,s}^{\tt GL}  \right)
.
\end{equation}

\begin{decomp}{Zhang--Shu classic convex decomposition}
In \cite{zhang2010,zhang2010b}, Zhang and Shu proposed the classic convex decomposition by using the tensor-product decomposition formulas (\ref{eq:GGL_x}) and (\ref{eq:GGL_y}):
\begin{align}\nonumber
\bar{u}_{i j}^{n} &
=
\kappa_1 \cdot \bar {u}_{ij}^{n} + \kappa_2 \cdot \bar {u}_{ij}^{n}
=
\kappa_1 \cdot \textrm{equation (\ref{eq:GGL_x})} +
\kappa_2 \cdot \textrm{equation (\ref{eq:GGL_y})} \\ \label{eq:411}
& =
\sum_{q=1}^Q \omega_{q}^{\tt G} \omega_1^{\tt GL} \left[
		 \kappa_1 u_{i-\frac{1}{2}, q}^{+} +
		 \kappa_1 u_{i+\frac{1}{2}, q}^{-} +
		 \kappa_2 u_{q, j-\frac{1}{2}}^{+} +
		 \kappa_2 u_{q, j+\frac{1}{2}}^{-}
\right]
 +
\sum_{s=2}^{L-1} \sum_{q=1}^{Q}  \omega_s^{\tt GL}  \omega_q^{\tt G}
\left[
\kappa_1  p_{i j}\left(x_{i,s}^{\tt GL}, y_{j,q}^{\tt G} \right) +
\kappa_2  p_{i j}\left(x_{i,q}^{\tt G},y_{j,s}^{\tt GL}  \right)
\right]
\end{align}
with
\begin{equation*}
	 \kappa_1 := \frac{\frac{a_{1}}{\Delta x}}{\frac{a_1}{\Delta x}+\frac{a_2}{\Delta y}}, \qquad
 \kappa_2 := \frac{\frac{a_{2}}{\Delta y}}{\frac{a_1}{\Delta x}+\frac{a_2}{\Delta y}}.
\end{equation*}
This classic convex decomposition has been widely used over the past decase.
\end{decomp}

\begin{decomp}{Jiang--Liu convex decomposition}
In \cite{jiang2018invariant}, Jiang and Liu used a simpler convex decomposition:
\begin{align}\nonumber
\bar{u}_{i j}^{n}
& =
\frac12 \cdot \bar {u}_{ij}^{n} + \frac12 \cdot \bar {u}_{ij}^{n}
=
\frac12 \cdot \textrm{equation (\ref{eq:GGL_x})} +
\frac12 \cdot \textrm{equation (\ref{eq:GGL_y})} \\ \label{eq:433}
& =
\sum_{q=1}^Q \frac{\omega_{q}^{\tt G}\omega_1^{\tt GL}}{2}\left[
		 u_{i-\frac{1}{2}, q}^{+} +
		 u_{i+\frac{1}{2}, q}^{-} +
		 u_{q, j-\frac{1}{2}}^{+} +
		 u_{q, j+\frac{1}{2}}^{-}
\right]
+
\sum_{s=2}^{L-1} \sum_{q=1}^{Q}  \frac{\omega_s^{\tt GL}  \omega_q^{\tt G} }{2}
\left[
 p_{i j}\left(x_{i,s}^{\tt GL}, y_{j,q}^{\tt G} \right) +
 p_{i j}\left(x_{i,q}^{\tt G},y_{j,s}^{\tt GL}  \right)
\right].
\end{align}
\end{decomp}

\begin{remark1}
Both the Zhang--Shu decomposition \eqref{eq:411} and the Jiang--Liu decomposition \eqref{eq:433} are examples of 2D feasible convex decomposition, and they share the same (classic) internal node set
\begin{equation}\label{eq:450}
	\mathbb{S}^{\tt Zhang-Shu}_{ij} = \bigcup_{s=2}^{L-1}\bigcup_{q=1}^{Q}\left\{(x_{i,s}^{\tt GL}, y_{j,q}^{\tt G} ),(x_{i,q}^{\tt G}, y_{j,s}^{\tt GL} )\right\}.
\end{equation}
\end{remark1}

\subsection{BP conditions via general convex decomposition}

Motivated by \cite{zhang2010,zhang2010b,jiang2018invariant}, this subsection
studies the BP conditions for the scheme (\ref{eq:171}) via the general feasible convex decomposition \eqref{2Ddecomp}.
One can rewrite \eqref{2Ddecomp} as
\begin{equation}\label{eq:454}
	\bar u_{ij}^{n} =
\sum_{q=1}^Q   \omega_q^{\tt G} \,
\widehat \omega_1 \left(  u_{i-\frac12,q}^+ +  u_{i+\frac12,q}^- \right)
+
\sum_{q=1}^Q   \omega_q^{\tt G} \,
\widehat \omega_2 \left(  u_{q,j-\frac12}^+ +  u_{q,j+\frac12}^- \right) + \Pi,
\end{equation}
where $\widehat \omega_1 := \min\{ \omega_1^-, \omega_1^+ \}$, $\widehat \omega_2 := \min\{ \omega_2^-, \omega_2^+ \}$, and
\begin{equation}\label{eq:Pi}
	\Pi :=
\sum_{q=1}^Q  \omega_q^{\tt G}
\left[
(\omega_1^- -\widehat \omega_1) u_{i-\frac12,q}^+ +
(\omega_1^+ -\widehat \omega_1) u_{i+\frac12,q}^-
+
(\omega_2^--\widehat \omega_2) u_{q,j-\frac12}^+ +
(\omega_2^+-\widehat \omega_2) u_{q,j+\frac12}^- \right]
+\sum_{s=1}^S \omega_s p_{ij}( x_{ij}^{(s)}, y_{ij}^{(s)}  ).
\end{equation}
By using a local scaling limiter \cite{zhang2010,zhang2010b}, one can enforce the DG solution polynomial $p_{ij}$ to satisfy the desired bounds on the boundary nodes:
\begin{equation}\label{eq:478a}
u_{i-\frac12,q}^{+}  \in G, \quad
u_{i+\frac12,q}^{-}  \in G, \quad
u_{q,j-\frac12}^{+}  \in G, \quad
u_{q,j+\frac12}^{-}  \in G, \quad
q = 1,\dots,Q, \quad \forall i,j,
\end{equation}
and
on the internal nodes:
\begin{equation}\label{eq:478b}
p_{ij}( x_{ij}^{(s)}, y_{ij}^{(s)}) \in G, \quad s = 1,\dots,S, \quad \forall i,j.
\end{equation}
Noting that (\ref{eq:Pi}) expresses $\Pi$ as a convex combination of the values in (\ref{eq:478a}) and (\ref{eq:478b}), we conclude that $\Pi \in G$, because $G$ is convex.
Substituting the decomposition \eqref{eq:454} into \eqref{eq:171}, one can rewrite the scheme \eqref{eq:171} as
\begin{equation}\label{eq:476}
		\bar {u}_{ij}^{n+1}  = \sum_{q=1}^Q    \omega_q^{\tt G}  \,
		\widehat \omega_1 (  H_{i+\frac12,q}^- + H_{i-\frac12,q}^+  ) +
		 \sum_{q=1}^Q    \omega_q^{\tt G}  \,
		\widehat \omega_2 ( H_{q,j+\frac12}^- + H_{q,j-\frac12}^+ ) + \Pi
\end{equation}
with
\begin{align*}
	H_{i+\frac12,q}^- &= u_{i+\frac12,q}^- - \frac{\Delta t}{\widehat \omega_1 \Delta x}
	\left( \hat f_1 \big( u_{i+\frac12,q}^-, u_{i+\frac12,q}^+ \big)
	- \hat f_1 \big( u_{i-\frac12,q}^+, u_{i+\frac12,q}^- \big)  \right),
	\\
	H_{i-\frac12,q}^+ & = u_{i-\frac12,q}^+ - \frac{\Delta t}{\widehat \omega_1 \Delta x}
	\left(   \hat f_1 \big( u_{i-\frac12,q}^+, u_{i+\frac12,q}^- \big)
	- \hat f_1 \big( u_{i-\frac12,q}^-, u_{i-\frac12,q}^+ \big)   \right),
	\\
	H_{q,j+\frac12}^- &= u_{q,j+\frac12}^- - \frac{\Delta t}{\widehat \omega_2 \Delta y}
	\left( \hat f_2 \big( u_{q,j+\frac12}^-, u_{q,j+\frac12}^+ \big)
	- \hat f_2 \big( u_{q,j-\frac12}^+, u_{q,j+\frac12}^- \big)  \right),
	\\
	H_{q,j-\frac12}^+ &= u_{q,j-\frac12}^+ - \frac{\Delta t}{\widehat \omega_2 \Delta y}
	\left(  \hat f_2 \big( u_{q,j-\frac12}^+, u_{q,j+\frac12}^- \big)
	- \hat f_2 \big( u_{q,j-\frac12}^-, u_{q,j-\frac12}^+ \big) \right),
\end{align*}
which are formally 1D three-point first-order schemes \eqref{1DBP} that satisfy
\begin{equation*}
	H_{i+\frac12,q}^-\in G, \quad  H_{i-\frac12,q}^+ \in G, \quad H_{q,j+\frac12}^- \in G, \quad H_{q,j-\frac12}^+ \in G,
\end{equation*}
under the CFL type conditions
\begin{equation}\label{eq:tempCFL}
a_1 {\Delta t}\le {\widehat \omega_1  \Delta x}, \quad a_2 {\Delta t}\le { \widehat \omega_2  \Delta y}.
\end{equation}
Because (\ref{eq:476}) is a convex combination form, by the convexity of $G$ we conclude that $\bar {u}_{ij}^{n+1} \in G$ under the conditions \eqref{eq:tempCFL}, which are equivalent to the following BP CFL condition \eqref{2Dhst_CFL_all}.
In summary, we arrive at the following theorem.

\begin{theorem}[BP via general convex decomposition]\label{thm:CFL}
If there is a 2D feasible convex decomposition in the form of \eqref{eq:QR2D} and the solution polynomial $p_{ij}$ satisfies (\ref{eq:478a}) and (\ref{eq:478b}) for all $i$ and $j$,
then the high-order scheme \eqref{eq:171} preserves
$\bar u_{ij}^{n+1}\in G$ under the BP CFL condition
\begin{equation}\label{2Dhst_CFL_all}
\Delta t \le c_0 \min \left\{ \frac{\omega_1^- \Delta x}{a_1}, \frac{\omega_1^+ \Delta x}{a_1} , \frac{\omega_2^- \Delta y}{a_2}, \frac{\omega_2^+ \Delta y}{a_2} \right\}.
\end{equation}
\end{theorem}

As direct consequences of
Theorem \ref{thm:CFL}, we have the following two corollaries.

\begin{corollary}[BP via Zhang--Shu convex decomposition]
If for all $i$ and $j$, the solution polynomial $p_{ij}$ satisfies (\ref{eq:478a}) and
$
p_{ij}(x,y) \in G$ for all $(x,y) \in \mathbb{S}^{\tt Zhang-Shu}_{ij}
$, then the high-order scheme \eqref{eq:171} preserves
$\bar u_{ij}^{n+1}\in G$ under the BP CFL condition
\begin{equation}\label{eq:CFL-ZS}
	\left( \frac{a_1}{\Delta x}+\frac{a_2}{\Delta y} \right)\Delta t \le \omega_1^{\tt GL}\,c_0
=\frac{c_0}{ L(L-1) } \quad \mbox{with} \quad L=\left \lceil \frac{k+3}{2} \right\rceil.
\end{equation}
\end{corollary}

\begin{corollary}[BP via Jiang--Liu convex decomposition]
If for all $i$ and $j$, the solution polynomial $p_{ij}$ satisfies (\ref{eq:478a}) and
$
p_{ij}(x,y) \in G$ for all $(x,y) \in \mathbb{S}^{\tt Zhang-Shu}_{ij}
$,
then the high-order scheme \eqref{eq:171} preserves
$\bar u_{ij}^{n+1}\in G$ under the BP CFL condition
\begin{equation*}
	2 \max \left\{\frac{a_1}{\Delta x},\frac{a_2}{\Delta y}\right\}\Delta t \le \omega_1^{\tt GL}\,c_0 =\frac{c_0}{ L(L-1) } \quad \mbox{with} \quad L=\left \lceil \frac{k+3}{2} \right\rceil.
\end{equation*}
\end{corollary}

\section{Optimal 2D convex decomposition for $\mathbb{P}^\textnormal{2}$ and $\mathbb{P}^\textnormal{3}$ on rectangular cells}\label{sec:3}

As we have seen, a convex decomposition like (\ref{2Ddecomp}) plays a critical role in constructing 2D high-order BP schemes: the choice of decomposition, in particular, the corresponding weights $\{\omega_1^-, \omega_1^+, \omega_2^-,  \omega_2^+\}$ affect the resulting BP CFL condition (\ref{2Dhst_CFL_all}). While the feasible convex decomposition approaches are not unique, it is natural to seek the {\em optimal convex decomposition} such that the resulting BP CFL condition \eqref{2Dhst_CFL_all} is mildest, i.e., it maximizes
\begin{equation*}
	\min  \left\{ \frac{\omega_1^- \Delta x}{a_1}, \frac{\omega_1^+ \Delta x}{a_1} , \frac{\omega_2^- \Delta y}{a_2}, \frac{\omega_2^+ \Delta y}{a_2} \right\}.
\end{equation*}

Seeking such an optimal convex decomposition in 2D is challenging.
We find that the classic Zhang--Shu decomposition
\eqref{eq:411} and the Jiang--Liu decomposition \eqref{eq:433} both are generally
not optimal for the $\mathbb P^k$ spaces.
Moreover, we discover the following novel convex decomposition on $\Omega_{ij}:=[x_{i-\frac12},x_{i+\frac12}]\times[y_{j-\frac12},y_{j+\frac12}]$ for $\mathbb{P}^2$ and $\mathbb{P}^3$.

\begin{decomp}{Optimal 2D convex decomposition}
	For $p_{ij}\in\mathbb{P}^2$ or $\mathbb P^3$,
	the cell average $\bar u_{ij}^n$ has the following convex decomposition
\begin{equation}\label{eq:QR2D}
\bar u_{ij}^n =
\frac{\mu_1}{2} \sum_{q=1}^Q \omega_{q}^{\tt G} \left[
  u_{i-\frac12,q}^{+} +
  u_{i+\frac12,q}^{-}
\right] +
\frac{\mu_2}{2} \sum_{q=1}^Q \omega_{q}^{\tt G} \left[
  u_{q,j-\frac12}^{+} +
  u_{q,j+\frac12}^{-}
\right] +
\omega\sum_{s} p_{ij}\left( \hat{x}_s,\hat{y}_s \right),
\end{equation}
with the internal nodes
\begin{equation}\label{eq:226}
\mathbb{S}_{ij}^{\tt optimal} = \{(\hat{x}_s,\hat{y}_s)\} =
\begin{dcases}
\; \left(
x_i,
y_j \pm \frac{\Delta y}{2\sqrt{3}}\sqrt{\frac{\phi_*-\phi_2}{\phi_*}}
\right),
& \text{if} \; \phi_1 \ge \phi_2, \\
\; \left(
x_i \pm \frac{\Delta x}{2\sqrt{3}}\sqrt{\frac{\phi_*-\phi_1}{\phi_*}} ,
y_j
\right),
& \text{if} \; \phi_1 < \phi_2,
\end{dcases}
\end{equation}
where
\begin{equation*}
	\phi_1 = \frac{ a_1}{\Delta x}, \quad
\phi_2 = \frac{a_2}{\Delta y}, \quad
\phi_* = \max\{\phi_1,\phi_2\}, \quad
\psi = \phi_1+\phi_2+2\phi_*, \quad
\mu_1 = \frac{\phi_1}{\psi}, \quad
\mu_2 = \frac{\phi_2}{\psi}, \quad
\omega = \frac{\phi_*}{\psi}.
\end{equation*}
\end{decomp}

It can be verified that the proposed 2D convex decomposition (\ref{eq:QR2D}) is feasible and optimal (see Theorem \ref{thm:OPT2DP2}) for both $\mathbb{P}^2$ and $\mathbb{P}^3$.
As a direct consequence of
Theorem \ref{thm:CFL}, we have following corollary.

\begin{corollary}[BP via optimal convex decomposition]
	If for all $i$ and $j$, the solution polynomial $p_{ij}$ satisfies (\ref{eq:478a}) and
$
p_{ij}(x,y) \in G$ for all $(x,y) \in \mathbb{S}^{\tt optimal}_{ij}
$, then the $\mathbb{P}^2$-based or $\mathbb{P}^3$-based high-order DG scheme \eqref{eq:171} preserves
	$\bar u_{ij}^{n+1}\in G$ under the BP CFL condition
\begin{equation}\label{eq:CFL2D}
\left[  2\frac{a_1}{\Delta x}+2\frac{a_2}{\Delta y}+ 4 \max\left\{
\frac{a_1}{\Delta x},\frac{a_2}{\Delta y}\right\} \right] \Delta t \le c_0.
\end{equation}
\end{corollary}

In Table \ref{tab:330}, we list and compare the corresponding BP CFL conditions and the internal node sets of decompositions (\ref{eq:QR2D}), (\ref{eq:411}) and (\ref{eq:433}) for $\mathbb{P}^2$ and $\mathbb{P}^3$. We observe that
our novel convex decomposition (\ref{eq:QR2D}) has some remarkable advantages, as summarized in the following remarks.

\begin{remark1}[Advantage in mildest BP CFL condition]\label{rem1}
One can observe from Table \ref{tab:330} that the proposed convex decomposition (\ref{eq:QR2D}) achieves a notably milder BP CFL condition than the existing ones, {\rm i.e.}, our BP CFL condition (\ref{eq:CFL2D}) is weaker than (\ref{eq:411}) and (\ref{eq:433}) respectively obtained via the Zhang--Shu and Jiang--Liu convex decompositions.  In fact, for the $\mathbb{P}^2$ or $\mathbb{P}^3$ space, no other 2D feasible convex decomposition can achieve an even milder BP CFL condition than (\ref{eq:CFL2D}), {\rm i.e.}, the proposed convex decomposition (\ref{eq:QR2D}) is {\em optimal}.
This will be theoretically proved in Theorem \ref{thm:OPT2DP2}.
\end{remark1}

\begin{remark1}[Advantage in fewer nodes]\label{rem2}
The internal node set $\mathbb{S}_{ij}^{\tt optimal}$ of the optimal convex decomposition (\ref{eq:QR2D}) contains only two nodes, which merge to a single node  $(x_i,y_j)$ in case of $\phi_1=\phi_2$. In comparison, the classic convex decomposition (\ref{eq:411}) or (\ref{eq:433}) needs much more internal nodes (approximately $2Q(L-2)$ in total). These two internal node sets, $\mathbb{S}_{ij}^{\tt Zhang-Shu}$ and $\mathbb{S}_{ij}^{\tt optimal}$, are shown in Figure \ref{fig:374} for further comparison. When the local scaling BP limiter is performed at the internal nodes in all computational cells, using the optimal convex decomposition (\ref{eq:QR2D}) reduces the computational cost of the BP limiting procedure.
\end{remark1}

\begin{remark1}[Easy implementation]
It is worth emphasizing that one
 only requires a slight and local modification to an existing code to enjoy the above-mentioned advantages of our optimal convex decomposition. Specifically, one only needs to replace the classic internal node set $\mathbb{S}_{ij}^{\tt Zhang-Shu}$ with the optimal internal node set $\mathbb{S}_{ij}^{\tt optimal}$
 in the BP limiting procedure, and then the theoretical BP CFL condition is improved to \eqref{eq:CFL2D}.
\end{remark1}

\begin{table}[h]
\centering
\caption{The theoretical BP CFL conditions and the internal node sets of the optimal convex decomposition (\ref{eq:QR2D}) and the convex decompositions (\ref{eq:411}) and (\ref{eq:433}) in 2D for the $\mathbb{P}^2$ and $\mathbb{P}^3$ spaces.}\label{tab:330}
\renewcommand\arraystretch{1.7}
\begin{tabular}{ lrrrr}
 \toprule[1.5pt]
  & BP CFL condition & BP CFL condition & Internal & Internal \\
 & general case & $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2} = h$ & $\mathbb{P}^2$ nodes & $\mathbb{P}^3$ nodes \\

 \midrule[1.5pt]

Optimal &
$\left[ 2 \frac{a_1}{\Delta x}+2 \frac{a_2}{\Delta y}+4 \max\left\{
\frac{a_1}{\Delta x},\frac{a_2}{\Delta y} \right\} \right] \Delta t \le c_0$
& $\Delta t \le \frac{c_0}{8} h$ & 1 $\sim$ 2 & 1 $\sim$ 2 \\
 Zhang \& Shu \cite{zhang2010} & $\left[6\frac{a_1}{\Delta x}+6\frac{a_2}{\Delta y}\right]\Delta t \le c_0$ & $\Delta t \le \frac{c_0}{12} h$ & 5 & 8 \\
 Jiang \& Liu \cite{jiang2018invariant} & $\left[12\max\left\{\frac{a_1}{\Delta x},\frac{a_2}{\Delta y} \right\}\right]\Delta t \le c_0$ & $\Delta t \le \frac{c_0}{12}h$ & 5 & 8\\

\bottomrule[1.5pt]
\end{tabular}
\end{table}

\begin{figure}
\centerline{
\begin{subfigure}{0.24\textwidth}
    \includegraphics[width=\textwidth]{figures/CDW_DG_P2_C2.png}
    \caption{Boundary nodes (black) and optimal internal nodes $\mathbb{S}_{ij}^{\tt optimal}$ (red) for $\mathbb{P}^2$.}

\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
    \includegraphics[width=\textwidth]{figures/CDW_DG_P3_C2.png}
    \caption{Boundary nodes (black) and optimal internal nodes $\mathbb{S}_{ij}^{\tt optimal}$ (red) for $\mathbb{P}^3$.}

\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
    \includegraphics[width=\textwidth]{figures/ZS_DG_P2_C2.png}
    \caption{Boundary nodes (black) and classic internal nodes $\mathbb{S}_{ij}^{\tt Zhang-Shu}$ (red) for $\mathbb{P}^2$.}

\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
    \includegraphics[width=\textwidth]{figures/ZS_DG_P3_C2.png}
    \caption{Boundary nodes (black) and classic internal nodes $\mathbb{S}_{ij}^{\tt Zhang-Shu}$ (red) for $\mathbb{P}^3$.}

\end{subfigure}
}
\caption{Nodes of the convex decompositions (\ref{eq:QR2D}) and the classic convex decomposition (\ref{eq:411}) on $\Omega_{ij} = [-1,1]^2$, for the $\mathbb{P}^2$ and $\mathbb{P}^3$ spaces, in the case of $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2}$. }
\label{fig:374}
\end{figure}

\begin{theorem}\label{thm:OPT2DP2}
	For both $\mathbb{P}^2$ and $\mathbb{P}^3$ spaces,
	the 2D convex decomposition (\ref{eq:QR2D}) is optimal among all feasible candidates.
\end{theorem}
\begin{proof}
	It can be easily verified that the 2D convex decomposition (\ref{eq:QR2D}) is feasible for $\mathbb{P}^2$ and $\mathbb{P}^3$. We will prove its optimality by contradiction. Assume that there is another feasible convex decomposition of the form
	\begin{equation}\label{eq:hyp}
		\bar {u}_{ij}^{n} =
		\sum_{q=1}^Q \omega_{q}^{\tt G}  \left[
		\omega_1^- u_{i-\frac{1}{2}, q}^{+} +
		\omega_1^+ u_{i+\frac{1}{2}, q}^{-} +
		\omega_2^- u_{q, j-\frac{1}{2}}^{+} +
		\omega_2^+ u_{q, j+\frac{1}{2}}^{-}
		\right]
		+ \sum_{s=1}^S \omega_s p_{ij} ( x_{ij}^{(s)}, y_{ij}^{(s)} ),
	\end{equation}
	which achieves a BP CFL condition milder than (\ref{eq:QR2D}), that is,
\begin{equation*}
		c_0 \min \left\{ \frac{\omega_1^- \Delta x}{a_1}, \frac{\omega_1^+ \Delta x}{a_1} , \frac{\omega_2^- \Delta y}{a_2}, \frac{\omega_2^+ \Delta y}{a_2} \right\}
	>
	c_0 \min \left\{
	\frac{\mu_1 \Delta x}{2 a_1},
	\frac{\mu_1 \Delta x}{2 a_1},
	\frac{\mu_2 \Delta y}{2 a_2},
	\frac{\mu_2 \Delta y}{2 a_2}
	\right\}
	=
	\frac{c_0}{2 \frac{a_1}{\Delta x}+2 \frac{a_2}{\Delta y}+4 \max\left\{
		\frac{a_1}{\Delta x},\frac{a_2}{\Delta y}\right\}}.
\end{equation*}
	In case of $\frac{a_1}{\Delta x} \ge \frac{a_2}{\Delta y}$, we have
	\begin{equation}\label{eq:541}
		\begin{aligned}
			1 & < \min \left\{
			\frac{\omega_1^- \Delta x}{a_1}, \frac{\omega_1^+ \Delta x}{a_1} ,
			\frac{\omega_2^- \Delta y}{a_2}, \frac{\omega_2^+ \Delta y}{a_2}
			\right\} \left[6 \frac{a_1}{\Delta x}+2 \frac{a_2}{\Delta y} \right] \\
			& \le
			3 \frac{\omega_1^- \Delta x}{a_1} \frac{a_1}{\Delta x}+
			3 \frac{\omega_1^+ \Delta x}{a_1} \frac{a_1}{\Delta x}+
			\frac{\omega_2^- \Delta y}{a_2} \frac{a_2}{\Delta y}+
			\frac{\omega_2^+ \Delta y}{a_2} \frac{a_2}{\Delta y}
			= 3(\omega_1^{+}+\omega_1^{-}) + (\omega_2^{+}+\omega_2^{-}).
		\end{aligned}
	\end{equation}
	In case of $\frac{a_1}{\Delta x} < \frac{a_2}{\Delta y}$, we have
	\begin{equation}\label{eq:556}
		\begin{aligned}
			1 & < \min \left\{
			\frac{\omega_1^- \Delta x}{a_1}, \frac{\omega_1^+ \Delta x}{a_1} ,
			\frac{\omega_2^- \Delta y}{a_2}, \frac{\omega_2^+ \Delta y}{a_2}
			\right\}
			\left[2 \frac{a_1}{\Delta x}+6 \frac{a_2}{\Delta y} \right] \\
			& \le
			\frac{\omega_1^- \Delta x}{a_1} \frac{a_1}{\Delta x}+
			\frac{\omega_1^+ \Delta x}{a_1} \frac{a_1}{\Delta x}+
			3 \frac{\omega_2^- \Delta y}{a_2} \frac{a_2}{\Delta y}+
			3 \frac{\omega_2^+ \Delta y}{a_2} \frac{a_2}{\Delta y}
			=(\omega_1^{+}+\omega_1^{-}) + 3(\omega_2^{+}+\omega_2^{-}).
		\end{aligned}
	\end{equation}
	No matter the hypothetical convex decomposition (\ref{eq:hyp}) is feasible for $\mathbb{P}^2$ or $\mathbb{P}^3$, it should hold exactly for $p_{ij}(x,y)=(x-x_i)^2$ and $p_{ij}(x,y)=(y-y_j)^2$. This gives
\begin{align*}
	\frac{\Delta x^2}{12} &=
(\omega_1^{+}+\omega_1^{-}) \frac{\Delta x^2}{4} +
(\omega_2^{+}+\omega_2^{-}) \frac{\Delta x^2}{12} +
\sum_{s=1}^N \omega_s ({x}_{ij}^{(s)}-x_i)^2,
\\
\frac{\Delta y^2}{12} &=
(\omega_1^{+}+\omega_1^{-}) \frac{\Delta y^2}{12} +
(\omega_2^{+}+\omega_2^{-}) \frac{\Delta y^2}{4} +
\sum_{s=1}^N \omega_s ({y}_{ij}^{(s)}-y_j)^2,
\end{align*}
	implying
	\begin{equation}\label{eq:553}
		3(\omega_1^{+}+\omega_1^{-}) + (\omega_2^{+}+\omega_2^{-}) \le 1
		\quad \text{and} \quad
		(\omega_1^{+}+\omega_1^{-}) + 3(\omega_2^{+}+\omega_2^{-}) \le 1,
	\end{equation}
	 which contradict with either (\ref{eq:541}) or (\ref{eq:556}). Hence the assumption is incorrect, and decomposition (\ref{eq:QR2D}) is optimal.
\end{proof}

\begin{remark1}
	The standard CFL condition for linear stability of the $\mathbb P^k$-based DG method with a $(k+1)$-stage $(k+1)$-order Runge--Kutta (RK) time discretization \cite{cockburn2001runge} is given by the following empirical formula
	\begin{equation}\label{eq:LS}
		\left(   \frac{a_1}{\Delta x}+ \frac{a_2}{\Delta y} \right) \Delta t \le \frac{1}{2k+1}.
	\end{equation}
	Table \ref{tab:specase} gives a comparison of different CFL conditions in the special case of $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2} = h$ for the $\mathbb P^2$-based (third-order) and  $\mathbb P^3$-based (fourth-order) DG methods.
	 One can see that if $c_0=1$,
	 the optimal BP CFL condition \eqref{eq:CFL2D} of the DG schemes (with the BP limiter) is even weaker than the standard one \eqref{eq:LS}.

  	\begin{table}[htbp]
  		\centering
  		\caption{Comparison of different CFL conditions in the special case of $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2} = h$.  }
  		\label{tab:specase}
  		\renewcommand\arraystretch{1.6}
  		\setlength{\tabcolsep}{6mm}{
  			\begin{tabular}{cccccc}

  				\toprule[1.5pt]

  				\multirow{2}{*}{} &
  				\multirow{2}{*}{linear stability} &
  				\multicolumn{2}{c}{ $c_0 = 1$ } &
  				\multicolumn{2}{c}{ $c_0 = {1}/{2}$} \\
  				\cmidrule(r){3-4} \cmidrule(l){5-6}
  				& &
  				optimal & classic &
  				optimal & classic \\

  				\midrule[1.5pt]

  				$\mathbb{P}^2$ & $\Delta t \leq \frac{1}{10}h$ &

  				\multirow{2}{*}{$\Delta t \leq \frac{1}{8}h$} &
  				\multirow{2}{*}{$\Delta t \leq \frac{1}{12}h$} &
  				\multirow{2}{*}{$\Delta t \leq \frac{1}{16}h$} &
  				\multirow{2}{*}{$\Delta t \leq \frac{1}{24}h$} \\

  				$\mathbb{P}^3$ & $\Delta t \leq \frac{1}{14}h$ &  & & & \\

  				\bottomrule[1.5pt]
  			\end{tabular}
  		}
  	\end{table}

\end{remark1}

\begin{remark1}
We would like to clarify that the optimal BP CFL condition \eqref{eq:CFL2D} is merely the best among all those achieved via feasible convex decomposition. It does not mean that such an optimal condition is always sharp or necessary, as other possible analysis approaches may give perhaps weaker BP conditions.
\end{remark1}

\section{Optimal 3D convex decomposition for $\mathbb{P}^\textnormal{2}$ and $\mathbb{P}^\textnormal{3}$ on cuboid cells}

The proposed 2D optimal convex decomposition (\ref{eq:QR2D}) can be extended to 3D and higher dimensions. We denote the maximum characteristic speeds in the $x$-, $y$- and $z$-directions by $a_1$, $a_2$, and $a_3$, respectively.
Let $\Omega_{ij\ell}=[x_{i-\frac12},x_{i+\frac12}]\times[y_{j-\frac12},y_{j+\frac12}] \times [z_{\ell-\frac12},z_{\ell+\frac12}]$ be a cuboid cell, with $\Delta x$, $\Delta y$, and $\Delta z$ denoting its lengths in the $x$-, $y$- and $z$-directions, respectively.

\begin{decomp}{Optimal 3D convex decomposition}
For $p_{ij\ell} \in \mathbb P^2$ or $\mathbb P^3$, the {\em optimal} 3D convex decomposition on $\Omega_{ij\ell}$ is given by
\begin{equation}\label{eq:QR3D}
\begin{aligned}
\bar u_{ij\ell}
& =
\frac{\mu_1}{2} \sum_{q=1}^{Q}\sum_{r=1}^{Q} \omega_{q}^{\tt G} \omega_{r}^{\tt G}
\left[
  u_{i-\frac12,q,r}^{+} +
  u_{i+\frac12,q,r}^{-}
\right] +
\frac{\mu_2}{2} \sum_{q=1}^{Q}\sum_{r=1}^{Q} \omega_{q}^{\tt G} \omega_{r}^{\tt G}
\left[
  u_{r,j-\frac12,q}^{+} +
  u_{r,j+\frac12,q}^{-}
\right]  \\
& +
\frac{\mu_3}{2} \sum_{q=1}^{Q}\sum_{r=1}^{Q} \omega_{q}^{\tt G} \omega_{r}^{\tt G}
\left[
  u_{q,r,\ell-\frac12}^{+} +
  u_{q,r,\ell+\frac12}^{-}
\right] +
\frac{\omega}{2}\sum_{s} p_{ij\ell}\left( \hat{x}_s,\hat{y}_s,\hat{z}_s \right),
\end{aligned}
\end{equation}
where $\bar u_{ij\ell}$ denotes the cell average of $p_{ij\ell}$ over $\Omega_{ij\ell}$, and
\begin{equation*}
	u_{i\pm\frac12,q,r}^{\mp} = p_{ij\ell} \big( x_{i\pm\frac12}  , y_{j,q}^{\tt G}, z_{\ell,r}^{\tt G} \big), \quad
u_{r,j\pm\frac12,q}^{\mp} = p_{ij\ell} \big( x_{i,r}^{\tt G}, y_{j\pm\frac12}  , z_{\ell,q}^{\tt G} \big), \quad
u_{q,r,k\pm\frac12}^{\mp} = p_{ij\ell} \big( x_{i,q}^{\tt G}, y_{j,r}^{\tt G}  , z_{\ell\pm\frac12}   \big).
\end{equation*}
In \eqref{eq:QR3D}, the weights $\mu_1$, $\mu_2$, $\mu_3$, and $\omega$ are given by
\begin{equation*}
	\mu_1 = \frac{\phi_1}{\psi}, \quad
\mu_2 = \frac{\phi_2}{\psi}, \quad
\mu_3 = \frac{\phi_3}{\psi}, \quad
\omega = \frac{\phi_*}{\psi}
\end{equation*}
with
\begin{equation*}
	\phi_1 = a_1/\Delta x, \quad
\phi_2 = a_2/\Delta y, \quad
\phi_3 = a_3/\Delta z, \quad
\psi = \phi_1+\phi_2+\phi_3+2\phi_*, \quad
\phi_* = \max \{\phi_1,\phi_2,\phi_3\},
\end{equation*}
and the internal nodes are given by
\begin{equation*}
\mathbb{S}_{ij\ell}^{\tt optimal} =	\left\{ \big( \hat{x}_{s},\hat{y}_{s},\hat{z}_{s} \big) \right\} =
\begin{dcases}
~ \left(
	x_i,
	y_j \pm \frac{\Delta y}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_2}{\phi_*}} ,
	z_\ell
\right)
\; \text{and} \;
\left(
	x_i,
	y_j,
	z_\ell \pm \frac{\Delta z}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_3}{\phi_*}}
\right),
& \text{if} \; \phi_1 = \max \{ \phi_1,\phi_2,\phi_3 \},\\
~
\left(
	x_i ,
	y_j ,
	z_\ell \pm \frac{\Delta z}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_3}{\phi_*}}
\right)
\; \text{and} \;
\left(
	x_i \pm \frac{\Delta x}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_1}{\phi_*}},
	y_j,
	z_\ell
\right),
& \text{if} \; \phi_2 = \max\{\phi_1,\phi_2,\phi_3 \},\\
~ \left(
	x_i \pm \frac{\Delta x}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_1}{\phi_*}},
	y_j,
	z_\ell
\right)
\; \text{and} \;
\left(
	x_i ,
	y_j \pm \frac{\Delta y}{\sqrt{6}} \sqrt{\frac{\phi_*-\phi_2}{\phi_*}},
	z_\ell
\right),
& \text{if} \; \phi_3 = \max\{\phi_1,\phi_2,\phi_3 \}.
\end{dcases}
\end{equation*}
\end{decomp}

\begin{theorem}\label{thm:OPT3DP2}
	For both $\mathbb{P}^2$ and $\mathbb{P}^3$ spaces,
the 3D convex decomposition (\ref{eq:QR3D}) is optimal among all feasible candidates.
\end{theorem}

The proof of Theorem \ref{thm:OPT3DP2} is similar to that of Theorem \ref{thm:OPT2DP2} and is thus omitted. As
summarized in Table \ref{tab:530},
the advantages of the optimal 3D convex decomposition (\ref{eq:QR3D}), over the 3D versions of the classic decompositions, are even greater than the 2D case.
Figure \ref{fig:627} shows
the boundary and internal nodes for further comparison.

\begin{table}[h]
\centering
\caption{BP CFL conditions and the internal node sets of the optimal 3D convex decomposition (\ref{eq:QR3D}) and the 3D versions of the Zhang--Shu and Jiang--Liu convex decompositions for the $\mathbb{P}^2$ and $\mathbb{P}^3$ spaces.}\label{tab:530}
\renewcommand\arraystretch{1.7}
\begin{tabular}{ lrrrr}
 \toprule[1.5pt]
  & BP CFL condition & BP CFL condition & Internal  & Internal  \\
 & general case & $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2} = \frac{\Delta z}{a_3} = h$ &  $\mathbb{P}^2$ nodes & $\mathbb{P}^3$ nodes  \\

 \midrule[1.5pt]

Optimal &
$\left[2\frac{a_1}{\Delta x}+2\frac{a_2}{\Delta y}+2\frac{a_3}{\Delta z}+4\max\left\{
\frac{a_1}{\Delta x},
\frac{a_2}{\Delta y},
\frac{a_3}{\Delta z}
\right\} \right] \Delta t \le c_0$
& $\Delta t \le \frac{c_0}{10} h$ & 1 $\sim$ 4 & 1 $\sim$ 4 \\
 Zhang \& Shu \cite{zhang2010} & $\left[6\frac{a_1}{\Delta x}+6\frac{a_2}{\Delta y}+6\frac{a_3}{\Delta z}\right]\Delta t \le c_0$ & $\Delta t \le \frac{c_0}{18} h$ & 19 & 48 \\
 Jiang \& Liu \cite{jiang2018invariant} & $\left[18 \max\left\{
\frac{a_1}{\Delta x},
\frac{a_2}{\Delta y},
\frac{a_3}{\Delta z}
\right\} \right]\Delta t \le c_0$ & $\Delta t \le \frac{c_0}{18} h$ & 19 & 48\\

\bottomrule[1.5pt]
\end{tabular}
\end{table}

\begin{figure}[htb]
	\centering
	\begin{subfigure}{0.24\textwidth}
		\includegraphics[width=\textwidth]{figures/CDW_DG_P2_C3.png}
		\caption{Boundary nodes (black) and optimal internal nodes $\mathbb{S}_{ij\ell}^{\tt optimal}$ (red) for $\mathbb{P}^2$.}

	\end{subfigure}
	\hfill
	\begin{subfigure}{0.24\textwidth}
		\includegraphics[width=\textwidth]{figures/CDW_DG_P3_C3.png}
		\caption{Boundary nodes (black) and optimal internal nodes $\mathbb{S}_{ij\ell}^{\tt optimal}$ (red) for $\mathbb{P}^3$.}

	\end{subfigure}
	\hfill
	\begin{subfigure}{0.24\textwidth}
		\includegraphics[width=\textwidth]{figures/ZS_DG_P2_C3.png}
		\caption{Boundary nodes (black) and classic internal nodes $\mathbb{S}_{ij\ell}^{\tt Zhang-Shu}$ (red) for $\mathbb{P}^2$.}

	\end{subfigure}
	\hfill
	\begin{subfigure}{0.24\textwidth}
		\includegraphics[width=\textwidth]{figures/ZS_DG_P3_C3.png}
		\caption{Boundary nodes (black) and classic internal nodes $\mathbb{S}_{ij\ell}^{\tt Zhang-Shu}$ (red) for $\mathbb{P}^3$.}

	\end{subfigure}
	\caption{Nodes of the optimal 3D decomposition (\ref{eq:QR3D}) and the 3D version of the classic Zhang--Shu decomposition on $\Omega_{ij\ell} = [-1,1]^3$, for $\mathbb{P}^2$ and $\mathbb{P}^3$, in the case of $\frac{\Delta x}{a_1} = \frac{\Delta y}{a_2} = \frac{\Delta z}{a_3}$.

		We only plot the nodes in the first octant $[0,1]^3$, while the other nodes are distributed symmetrically.}
	\label{fig:627}
\end{figure}

\begin{figure}[h]
	\centering
	\centerline{
		\hfill

		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P2-RK-L1.pdf}

		\hfill
		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P2-RK-L2.pdf}
		\hfill

		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P2-RK-Linf.pdf}

		\hfill
	}

	%

	\centerline{
		\hfill

		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P3-RK-L1.pdf}

		\hfill
		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P3-RK-L2.pdf}
		\hfill

		\includegraphics[width=0.32\textwidth]{result/linearLongTime/P3-RK-Linf.pdf}

		\hfill
	}
	\caption{Example \ref{Ex1}: Time evolution of the $L^1$, $L^2$, and $L^{\infty}$ errors in the numerical solutions obtained using the third-order ($\mathbb{P}^2$; top row) and fourth-order ($\mathbb{P}^3$; bottom row) BP DG schemes designed via different approaches and by using different time step-sizes.

	}
	\label{fig:ex1_2}
\end{figure}

\section{Numerical experiments}

In this section, we test the accuracy, robustness, and efficiency of the high-order BP DG schemes designed via the proposed optimal convex decomposition \eqref{eq:QR2D}, which are referred to as the ``{\tt optimal} approach'' for short. For comparison, we also consider the high-order BP DG schemes designed via the classic Zhang--Shu convex decomposition \eqref{eq:411}, which are referred to as the ``{\tt classic} approach'' for short.
While the convex decomposition is independent of the choice of BP numerical fluxes, we adopt
the global Lax--Friedrichs flux with $c_0=1$ in all the presented tests.
Unless otherwise stated, we employ the three-stage third-order SSP RK discretization  \cite{GottliebKetchesonShu2011} (abbreviated as {\tt SSPRK3}) for the $\mathbb P^2$-based (third-order) DG scheme,
and  the five-stage fourth-order SSP RK time discretization \cite{GottliebKetchesonShu2011} (abbreviated as {\tt SSPRK4})
 for the $\mathbb P^3$-based (fourth-order) DG scheme.
The SSP coefficients for {\tt SSPRK3} and {\tt SSPRK4} are $C_{\tt SSP}=1$ and $C_{\tt SSP} \approx 1.508$, respectively.
All  the  methods  are  implemented using C++ language with double precision on a Linux server with
Intel(R) Xeon(R) Platinum 8268 CPU @ 2.90GHz 2TB RAM.

\begin{example}[Linear convection equation]\label{Ex1}
	We start with the two-dimensional linear convection equation
	\begin{equation}\label{eq:791}
		u_t+u_x + u_y=0, \qquad (x,y,t)\in[-1,1]\times[-1,1]\times\mathbb{R}^+,
	\end{equation}
	with periodic boundary conditions and the initial data $u(x,y,0) = \sin(\pi(x+y))$.
	The exact solution satisfies a maximum principle, implying the convex invariant region $G=[-1,1]$.
	We simulate this problem until $t=50$ to study the long-time stability of the BP DG schemes with the BP limiter on the  uniform mesh of $100 \times 100$ cells.
	Figure \ref{fig:ex1_2} shows the time evolution of the numerical errors in the
	$L^1$, $L^2$, and $L^{\infty}$ norms, respectively. In the simulations, we adopt three different time step-sizes,
	namely, $\tau_0 :=  C_{\tt SSP}  \tau^{\tt BP}_{\tt O}$ with $\tau^{\tt BP}_{\tt O}$ being
	the maximum $\Delta t$ determined by the optimal BP CFL condition \eqref{eq:CFL2D},
	$\tau_1 :=  C_{\tt SSP}  \tau^{\tt BP}_{\tt C}$ with $\tau^{\tt BP}_{\tt C}$ being
	the maximum $\Delta t$ determined by the classic Zhang--Shu BP CFL condition \eqref{eq:CFL-ZS},
	and $\tau_2 :=  C_{\tt SSP}  \tau^{\tt LS}$ with $\tau^{\tt LS}$ being the
	maximum $\Delta t$ determined by the standard CFL condition \eqref{eq:LS} for linear stability.
	The CPU time of all these simulations is presented in Table \ref{tab:CPUEx1}.
	One can see that all the errors shown in Figure \ref{fig:ex1_2} do not exhibit any sign of exponential growth, which indicates the simulations are stable.
	We also observe that, when the identical time step is used,
	the errors of the {\tt optimal} approach is smaller or comparable to those of the {\tt classic} one,
	but the {\tt optimal} approach uses less CPU time (see Table \ref{tab:CPUEx1}) due to the fewer internal nodes.
	It is seen that the numerical errors are slightly larger, if a bigger time step is adopted, as expected.
	The {\tt optimal} approach allows a larger time step, with which the CPU time is much less, as shown in Table \ref{tab:CPUEx1}.

  	\begin{table}[htbp]
	\centering
	\caption{CPU time in seconds for Example \ref{Ex1}.}
		\label{tab:CPUEx1}
	\renewcommand\arraystretch{1.5}
	\setlength{\tabcolsep}{5mm}{
		\begin{tabular}{cccccc}

			\toprule[1.5pt]

			\multirow{2}{*}{} &
			\multicolumn{3}{c}{{\tt optimal} approach} &
			\multicolumn{2}{c}{{\tt classic} approach} \\
			\cmidrule(r){2-4} \cmidrule(l){5-6}
			&
			$\tau_0$ & $\tau_1$ & $\tau_2$ &
			$\tau_1$ & $\tau_2$  \\

			\midrule[1.5pt]

			$\mathbb{P}^2$ & 1563.22 & 2410.99 & 1980.90 & 2603.67 & 2166.29 \\

			$\mathbb{P}^3$ & 3058.73 & 4564.84 & 5330.88 & 4947.13 & 5775.08 \\

			\bottomrule[1.5pt]
		\end{tabular}
	}
\end{table}

\end{example}

\iffalse
\begin{example}[Burgers' equation with smooth initial condition]\label{Ex2}
In order to examine the convergence,
this example considers the inviscid Burgers' equation
\begin{align}\label{eq:Burgers}
	u_t+(\frac{u^2}{2})_x + (\frac{u^2}{2})_y=0, \qquad (x,y,t)\in[-1,1]\times[-1,1]\times\mathbb{R}^+,
\end{align}
with periodic boundary conditions and the smooth initial data $u(x,y,0) = \sin(\pi(x+y))$.
The exact solution also obeys the maximum principle with $G=[-1,1]$.
The exact solution is smooth up to $t = \frac{1}{2\pi}$. {\color{blue}Later, an emerged moving shock starts interacting with a rarefaction wave.}
Table \ref{tab:ex2_1} lists the $L^1$ and $L^{\infty}$ errors as well as the corresponding convergence rates for the numerical solutions at $ t = 0.05 $
obtained by  the {\tt optimal} and {\tt classic} approaches with the same time step-size $\Delta t = C_{\tt SSP}  {\Delta x}/12 $ for the sake of comparing numerical errors.

the BP schemes designed via
{\tt optimal-$\tau_2$} and {\tt  classic-$\tau_2$}

At $ t = 0.05 $, the solution is still smooth. We list in Table \ref{tab:ex2_1} the $L^1$ errors of the numerical obtained by {\tt optimal-$\tau_2$} and {\tt  classic-$\tau_2$} approaches. The results confirm that the designed order of accuracy has been maintained. However, the accuracy degenerates for the Runge-Kutta time discretization with $\mathbb{P}^3$-based. Similarly to the Example 1a, the multi-step method can be used as a remedy. When the CPU time is compared, the {\tt optimal} approach achieves similar level of $L^1$ error with smaller CPU time. For fourth-order methods (based on $\mathbb{P}^3$), the time steps are identical between {\tt optimal-$\tau_2$} and {\tt  classic-$\tau_2$} methods, the CPU time saved is due to the smaller number of nodes in proposed optimal convex decomposition. For third-order methods (based on $\mathbb{P}^2$), the time steps of {\tt optimal-$\tau_2$} methods is one fifth larger, leading to a greater efficiency advantage.

	In Table \ref{tab:ex1_1}, we include the errors and the corresponding rate of convergence of the numerical solutions at $t=0.1$. We can observe the accuracy degeneracy in $L^{\infty}$ norm for the SSP-RK methods. This is due to the lower order accuracy in the intermediate stages of the Runge–Kutta method. Such a phenomenon is observed in the original work of Zhang and Shu \cite{zhang2010,zhang2010b}, also has been pointed out in XXXX. One can instead use SSP multi-step (MS) methods for time integration.  We demonstrate the rate of convergence using SSP multi-step methods. The third- and fourth-order of accuracy of the BP schemes, using either classic or proposed optimal convex decomposition, are verified.

\begin{table}[htbp]
	\centering
	\caption{ Example 2 (Burgers' equation with smooth initial condition). The above table lists $L^1$ errors, together with the corresponding order of convergence and CPU times. The numerical results are obtained using third- and fourth-order BP DG schemes based on polynomial spaces $\mathbb{P}^2$ and $\mathbb{P}^3$. We use time integration methods SSP-RK and SSP-MS. The time steps are chosen based on appoaches {\tt optimal-$\tau_1$} and {\tt classic-$\tau_1$}.  $t=0.05$, $ \Delta x = \Delta y = 2/N $.
	}
	\label{tab:ex2_1}
	\setlength{\tabcolsep}{2mm}{
		\begin{tabular}{cllclclclc}

			\toprule[1.5pt]
			\multirow{2}{*}{ } &
			\multirow{2}{*}{$ N$} &
			\multicolumn{4}{c}{{\tt optimal} approach} &
			\multicolumn{4}{c}{{\tt classic} approach} \\
			\cmidrule(r){3-6} \cmidrule(l){7-10}
			&
			& $ L^{1} $ error & Order & $ L^{\infty} $ error & Order & $ L^{1} $ error & Order & $ L^{\infty} $ error & Order \\

			\midrule[1.5pt]

			\multirow{6}{*}{ $\mathbb{P}^2$-RK3 }
			& 20 & 5.20e-4 & -    & 3.35e-3 & -    & 5.37e-4 & -    & 3.35e-3 & -    \\
			& 40 & 7.03e-5 & 2.89 & 5.37e-4 & 2.64 & 7.40e-5 & 2.86 & 5.99e-4 & 2.49 \\
			& 80 & 1.01e-5 & 2.80 & 1.24e-4 & 2.12 & 1.05e-5 & 2.82 & 1.36e-4 & 2.14 \\
			& 160& 1.45e-6 & 2.80 & 2.92e-5 & 2.08 & 1.51e-6 & 2.80 & 3.16e-5 & 2.11 \\
			& 320& 2.10e-7 & 2.79 & 7.08e-6 & 2.05 & 2.19e-7 & 2.79 & 7.56e-6 & 2.06 \\
			& 640& 3.03e-8 & 2.80 & 1.75e-6 & 2.01 & 3.12e-8 & 2.81 & 1.84e-6 & 2.04 \\

			\midrule[1.5pt]

			\multirow{6}{*}{ $\mathbb{P}^2$-MS3 }
			& 20 & 4.88e-4 & -    & 3.33e-3 & -    & 4.91e-4 & -    & 3.33e-3 & -    \\
			& 40 & 6.07e-5 & 3.01 & 4.66e-4 & 2.84 & 6.10e-5 & 3.01 & 4.66e-4 & 2.84 \\
			& 80 & 7.76e-6 & 2.97 & 6.50e-5 & 2.84 & 7.78e-6 & 2.97 & 6.50e-5 & 2.84 \\
			& 160& 1.00e-6 & 2.95 & 9.70e-6 & 2.74 & 1.01e-6 & 2.95 & 9.70e-6 & 2.74 \\
			& 320& 1.32e-7 & 2.93 & 1.45e-6 & 2.74 & 1.32e-7 & 2.93 & 1.45e-6 & 2.74 \\
			& 640& 1.75e-8 & 2.92 & 2.19e-7 & 2.73 & 1.75e-8 & 2.92 & 2.19e-7 & 2.73 \\

			\midrule[1.5pt]

			\multirow{6}{*}{ $\mathbb{P}^3$-RK4}
			& 10 & 5.73e-4 & -    & 6.11e-3 & -    & 6.87e-4 & -    & 6.42e-3 & -    \\
			& 20 & 7.31e-5 & 2.97 & 5.47e-4 & 3.48 & 8.57e-5 & 3.00 & 5.85e-4 & 3.46 \\
			& 40 & 9.70e-6 & 2.91 & 1.26e-4 & 2.12 & 1.11e-5 & 2.95 & 1.76e-4 & 1.73 \\
			& 80 & 1.37e-6 & 2.82 & 3.51e-5 & 1.85 & 1.41e-6 & 2.97 & 3.97e-5 & 2.15 \\
			& 160& 1.84e-7 & 2.90 & 7.85e-6 & 2.16 & 1.80e-7 & 2.97 & 8.27e-6 & 2.26 \\
			& 320& 2.73e-8 & 2.75 & 2.21e-6 & 1.83 & 2.80e-8 & 2.68 & 2.32e-6 & 1.83 \\

			\midrule[1.5pt]

			\multirow{6}{*}{ $\mathbb{P}^3$-MS4}
			& 10 & 4.95e-4  & -    & 5.47e-3 & -    & 5.35e-4  & -    & 5.58e-3 & -    \\
			& 20 & 3.91e-5  & 3.66 & 5.26e-4 & 3.38 & 4.09e-5  & 3.71 & 5.26e-4 & 3.41 \\
			& 40 & 2.56e-6  & 3.93 & 3.84e-5 & 3.78 & 2.63e-6  & 3.96 & 3.84e-5 & 3.78 \\
			& 80 & 1.59e-7  & 4.01 & 2.86e-6 & 3.75 & 1.61e-7  & 4.03 & 2.86e-6 & 3.75 \\
			& 160& 1.01e-8  & 3.98 & 1.93e-7 & 3.89 & 1.02e-8  & 3.98 & 1.93e-7 & 3.89 \\
			& 320& 6.41e-10 & 3.98 & 1.26e-8 & 3.94 & 6.41e-10 & 3.99 & 1.26e-8 & 3.94 \\

			\midrule[1.5pt]

			\multirow{4}{*}{ $\mathbb{P}^3$-RK3}
			& 10 & 5.52e-4 & -    & 6.08e-3 & -    & 6.22e-4 & -    & 6.22e-3 & -    \\
			& 20 & 4.82e-5 & 3.52 & 5.28e-4 & 3.53 & 5.33e-5 & 3.54 & 5.25e-4 & 3.57 \\
			& 40 & 4.23e-6 & 3.51 & 3.86e-5 & 3.77 & 4.46e-6 & 3.58 & 4.18e-5 & 3.65 \\
			\multirow{1}{*}{$\Delta t = \frac{1}{14}\Delta x^{\frac{4}{3}}$}
			& 80 & 3.46e-7 & 3.61 & 5.88e-6 & 2.71 & 3.65e-7 & 3.61 & 6.48e-6 & 2.69 \\
			& 160& 2.81e-8 & 3.62 & 9.46e-7 & 2.63 & 2.95e-8 & 3.63 & 9.99e-7 & 2.70 \\
			& 320& 2.13e-9 & 3.72 & 1.36e-7 & 2.80 & 2.19e-9 & 3.75 & 1.34e-7 & 2.90 \\

			\bottomrule[1.5pt]
		\end{tabular}
	}
\end{table}

\end{example}

\fi

\begin{example}[Riemann problem of Burgers' equation]\label{Ex3}
This example \cite{lu2021oscillation} considers	the inviscid Burgers' equation
	\begin{align
		u_t+\bigg(\frac{u^2}{2}\bigg)_x + \bigg(\frac{u^2}{2}\bigg)_y=0, \qquad (x,y,t)\in[0,1]\times[0,1]\times\mathbb{R}^+,
	\end{align}
	with outflow boundary conditions and the initial condition
\begin{align*}
	u(x,y,0) =
	\begin{cases}
		-0.2, & \text{ if } x < 0.5, y> 0.5, \\
		-1, & \text{ if } x > 0.5, y> 0.5, \\
		0.5, & \text{ if } x < 0.5, y < 0.5, \\
		0.8, & \text{ if } x > 0.5, y < 0.5.
	\end{cases}
\end{align*}
The exact solution  of this example also obeys the maximum principle with $G=[-1,0.8]$.
Figure \ref{fig:ex3_1} displays the numerical results at $t=0.5$ computed with $256\times 256$ uniform cells.
We observe that the {\tt optimal} approach with time step-size $\tau_0=C_{\tt SSP}  \tau^{\tt BP}_{\tt O}$ and the {\tt classic} approach with time step-size $\tau_1=C_{\tt SSP}  \tau^{\tt BP}_{\tt C}$ give very similar results, and
the discontinuities are equally well resolved by both approaches.
However, the CPU time of the {\tt optimal} approach is much less than that of the
{\tt classic} approach, as exhibited in Table \ref{tab:ex-cpu}.

\begin{figure}[h]
	\centering
	\begin{subfigure}{0.31\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P2-NoWENO-CWDBig-2D.jpeg}
		\caption{ $\mathbb{P}^2$: {\tt optimal} approach.}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.31\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P2-NoWENO-ZhangShuNormal-2D.jpeg}
		\caption{ $\mathbb{P}^2$: {\tt classic} approach.}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.35\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P2-NoWENO-diagonal-Big.pdf}
		\caption{ $\mathbb{P}^2$: cut along the diagonal line  $x=1-y$.}
	\end{subfigure}

	\begin{subfigure}{0.31\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P3-NoWENO-CWDBig-2D.jpeg}
		\caption{ $\mathbb{P}^3$: {\tt optimal} approach.}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.31\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P3-NoWENO-ZhangShuBig-2D.jpeg}
		\caption{$\mathbb{P}^3$: {\tt classic} approach.}
	\end{subfigure}
	\hfill
	\begin{subfigure}{0.35\textwidth}
		\includegraphics[width=\textwidth]{result/Burgers-Riemann/P3-NoWENO-diagonal-Big.pdf}
		\caption{$\mathbb{P}^3$: cut along the diagonal line  $x=1-y$.}
	\end{subfigure}

	\caption{Example \ref{Ex3}: The numerical solutions at $t=0.5$ obtained via the {\tt optimal} approach and the {\tt classic} approach.
	}
	\label{fig:ex3_1}
\end{figure}

\end{example}

\begin{table}[htb]
	\centering
	\caption{CPU time in seconds for Examples \ref{Ex3}--\ref{Ex5}.}
	\label{tab:ex-cpu}
	\setlength{\tabcolsep}{5.6mm}{
		\begin{tabular}{lccccc}

			\toprule[1.5pt]

			& \multirow{2}{*}{Approach}

			& Example \ref{Ex3}  & Example \ref{Ex4}      & Example \ref{Ex5} \\

			& & 2D Riemann problem  &  Mach 80 jet & Mach 2000 jet \\

			\midrule[1.5pt]
			\multirow{2}{*}{ $\mathbb{P}^2$}
			& optimal-$\tau_0$   & 309.39  & 13297.97 & 10708.38 \\
			& classic-$\tau_1$   & 504.95 & 20501.56 & 16527.88 \\

			\midrule[1.5pt]
			\multirow{2}{*}{ $\mathbb{P}^3$}
			& optimal-$\tau_0$   & 664.71 & 30861.78 & 24304.09 \\
			& classic-$\tau_1$   & 1106.00& 42714.91 & 34896.86 \\

			\bottomrule[1.5pt]
		\end{tabular}
	}
\end{table}

\begin{example}[Mach 80 jet problem of Euler equations]\label{Ex4}
	This example simulates a Mach 80 jet \cite{ha2005numerical,zhang2010b,liu2022essentially} by solving the two-dimensional Euler equations,
	which can be
	written in the form of (\ref{2DCL}) with
	\begin{equation} \label{Eq:Euler}

			u=\begin{pmatrix}
				\rho \\
				\rho v_1 \\
				\rho v_2 \\
				E
			\end{pmatrix}, \qquad {f_1}(u)=\begin{pmatrix}
				\rho v_1 \\
				\rho {v_1}^{2}+p \\
				\rho v_1 v_2 \\
				(E+p) v_1
			\end{pmatrix}, \qquad {f_2}(u)=\begin{pmatrix}
				\rho v_2 \\
				\rho v_1 v_2 \\
				\rho {v_2}^{2}+p \\
				(E+p) v_2
		\end{pmatrix}

	\end{equation}
with $E=\frac{1}{2} \rho ( {v_1}^{2} + {v_2}^{2})+\rho e$ and $p=(\gamma-1) \rho e$.
Here, $\rho$ is the density, $(v_1,v_2)$ denotes the velocity, $p$ is the pressure, $E$ is the total energy, and $e$ is the specific internal energy. The ratio of specific heats $\gamma$ is set to be $5/3$.
The density and pressure should be positive, yielding the invariant region
$
G=\{ u: \rho>0, p>0 \},
$ which is a convex set \cite{zhang2010b}.

Initially,
the computation domain $[0,2]\times[-0.5,0.5]$ is  full of the ambient gas with $(\rho,v_1,v_2,p) = (5,0,0,0.4127)$.
The jet with state $(\rho,v_1,v_2,p) = (5,30,0,0.4127)$ is injected into the domain from the left boundary between $y=-0.05$ and $0.05$. All the other boundaries are set as  outflow boundary conditions, as in \cite{ha2005numerical,liu2022essentially}.
This is a benchmark yet challenging test, and a high-order numerical scheme without any BP techniques may easily produce negative density and/or negative pressure, which
eventually causes the breakdown of the simulation code.
We perform the simulation until $t=0.07$ on the uniform mesh of $480\times 240$ cells.
The numerical results of density are presented in Figure \ref{fig:ex4_1}, from which we
	clearly observe that the critical features of the jet: cocoons, bow shock, shear flows, etc.
	All these features
	are well captured by the BP DG methods and agree with those presented in \cite{ha2005numerical,liu2022essentially}.
	The results of the {\tt optimal}
	approach with time step $\tau_0=C_{\tt SSP}  \tau^{\tt BP}_{\tt O}$
	are comparable  to those of
	 the {\tt classic} approach with time step $\tau_1=C_{\tt SSP}  \tau^{\tt BP}_{\tt C}$.
	As shown in Table \ref{tab:ex-cpu}, the {\tt optimal} approach  allows a larger time step and
	takes much less CPU time than the {\tt classic} approach.

For both approaches,
the local scaling BP limiter \cite{zhang2010b} is necessary to enforce the conditions (\ref{eq:478a}) and (\ref{eq:478b}). Due to the presence of strong shocks in this and next examples, the WENO limiter \cite{qiu2005runge} is also used, right before the BP limiter, within some adaptively detected troubled cells to suppress potential numerical oscillations.

\begin{figure}[h]
	\centering
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet80/P2-4-New-Big/P2-v2-density.jpeg}
			\caption{$\mathbb{P}^2$: {\tt optimal} approach.}
		\end{center}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet80/P2-2-ZhangShu-Normal/P2-v2-density.jpeg}
			\caption{$\mathbb{P}^2$: {\tt classic} approach.}
		\end{center}
	\end{subfigure}

	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet80/P3-5-New-Big/P3-vv1-density.jpeg}
			\caption{$\mathbb{P}^3$: {\tt optimal} approach.}
		\end{center}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet80/P3-4-ZhangShu-Big/P3-vv1-density.jpeg}
			\caption{$\mathbb{P}^3$: {\tt classic} approach.}
		\end{center}
	\end{subfigure}

	\caption{Example \ref{Ex4}: The numerical density at $t=0.07$ obtained by using the third-order (top row)
		and the fourth-order (bottom row) BP DG schemes designed via the {\tt optimal} approach and the {\tt classic} approach.}
	\label{fig:ex4_1}
\end{figure}

\begin{figure}[h]
	\centering
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet2000/P2-4-New-Big/P2-v2-density.jpeg}
			\caption{$\mathbb{P}^2$: {\tt optimal} approach.}
		\end{center}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet2000/P2-2-ZhangShu-Normal/P2-v2-density.jpeg}
			\caption{$\mathbb{P}^2$: {\tt classic} approach.}
		\end{center}
	\end{subfigure}

	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet2000/P3-5-New-Big/P3-vv2-density.jpeg}
			\caption{$\mathbb{P}^3$: {\tt optimal} approach.}
		\end{center}
	\end{subfigure}
	\hfill
	\begin{subfigure}[b]{0.49\textwidth}
		\begin{center}
			\includegraphics[width=\textwidth]{result/Jet2000/P3-4-ZhangShu-Big/P3-vv2-density.jpeg}
			\caption{$\mathbb{P}^3$: {\tt classic} approach.}
		\end{center}
	\end{subfigure}

	\caption{Same as Figure \ref{fig:ex4_1} except for Example \ref{Ex5} (Mach 2000 jet) at $t=0.001$.}
	\label{fig:ex4_2}
\end{figure}

\end{example}

\begin{example}[Mach 2000 jet problem of Euler equations]\label{Ex5}
Last, a Mach 2000 jet is considered with the Euler equation (\ref{Eq:Euler}) in the domain $[0,1]\times[-0.25,0.25]$.
The setup is the same as Example \ref{Ex4}, except the jet state fixed as $(\rho,v_1,v_2,p) = (5,800,0,0.4127)$ for  $y\in[-0.05,0.05]$ on the left boundary ($x=0$).
All the other boundaries are of outflow conditions.
The much higher Mach number renders this jet test more challenging than Example \ref{Ex4}.
The simulation is performed until $t=0.001$ with $480\times 240$ cells.
Figure \ref{fig:ex4_2} presents the numerical results of density, which demonstrate the comparably high resolution and excellent robustness for both the {\tt optimal} approach and the {\tt classic} approach.
Table \ref{tab:ex-cpu} also displays the CPU time in this test for both approaches, further confirming the notable advantage of the {\tt optimal} approach in efficiency.

\end{example}

\section{Summary}

In this paper, we proposed the problem of seeking the optimal
convex decomposition of the cell average for
 constructing high-order BP schemes of hyperbolic conservation laws in multiple dimensions within the Zhang--Shu framework.
It was observed that the classic Zhang--Shu convex decomposition, based on the tensor product of Gauss--Lobatto and Gauss quadratures, is generally not optimal in the multidimensional cases.
For the $\mathbb P^2$ and $\mathbb P^3$ spaces, which are typically used in the third-order and fourth-order DG schemes, we discovered the optimal convex decomposition that achieves the mildest BP CFL condition yet requires much fewer internal nodes.
Based on our optimal convex decomposition, we established more efficient high-order BP schemes, which allow a larger BP time step-size than the classic one, in the Zhang--Shu framework.
We presented several numerical examples to validate the remarkable advantages of using our optimal decomposition over the classic one in terms of efficiency.

The discovery of the optimal convex decomposition was quite nontrivial and might have a broad impact, as it would
lead to an overall improvement of third-order and fourth-order BP schemes for
a large class of hyperbolic or convection-dominated equations at the cost of only a slight and local modification to the implementation code.
Our work in this paper was limited to the multivariate polynomial spaces $\mathbb{P}^2$ and $\mathbb{P}^3$. In more general cases,
many questions about the optimal convex decomposition are yet open; for example,
what are the optimal decompositions for more general polynomial spaces $\mathbb{P}^k$ with $k\ge 4$ on Cartesian meshes,  triangular meshes, and more general unstructured meshes?
We
hope this paper could motivate further exploration along this direction in the future.

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Lincoln & York are one of the UK's leading private label coffee roasters, supplying over 300 customers in 10 countries across Europe. Established in 1994, we have been sourcing, roasting and packaging coffee from over 40 origins to deliver completely bespoke products to customers for 25 years.
As one of the UK's largest coffee roasters, we are the discrete manufacturers behind hundreds of coffee blends across the UK and Europe, working with customers to provide products tailored to commercial requirements and the constantly evolving consumer trends.
At Lincoln & York we are constantly striving to advance and champion innovation and heritage in the coffee industry, through bringing exciting new blends and processes to the forefront of the sector.
Ultimately, our mission is to become the UK coffee roaster of choice for the out of home market and be the face and voice of private label coffee.
Our fruitful growth within the coffee industry allows us to foster a working environment that stimulates development of our employees, whilst we continue to find new and improved ways of sustainably sourcing our coffee.
As coffee continues to be the world's drink of choice, our passion for roasting coffee will continue driving us to master each individual blend to perfection.


LITTLE ROCK, Ark. — Zips Car Wash opened a new location in Andover, Kansas.
LITTLE ROCK, Ark. — Zips Car Wash is offering more convenience for Wichita area residents with a new location in Andover, Kansas, just east of Wichita, according to a press release.
This new store, located at 565 S. Andover Rd., will allow the company to serve customers at seven stores within a 25-mile radius with Unlimited Wash Clubs and always-free self-serve vacuums — two features Zips is known for across the nation, the release continued.
Zips Car Wash purchased the land and built its newest carwash from the ground up, the release noted.
Features include Zips Unlimited Wash Club packages, which customers can use at any Zips Car Wash location; quick, convenient services; and free self-serve vacuums, the release stated.
Due to a city ordinance, Zips Car Wash in Andover will be open from 7 a.m. to 7 p.m. daily with the exception of Christmas day, the release stated; all other Zips sites operate from 8 a.m. to 8 p.m. daily.
The growing carwash chain now serves customers at 137 locations across the country, the release concluded.


The fall semester writing class recently completed Quiz #1. Here are the questions and answers, which can help ESL students learn English writing skills.
Greg Brown was A high school student. While driving to school, he had A motorcycle accident. At 8:45 AM. he was driving down _X Queen Street. At THE intersection, THE light turned red. He tried to stop but fell down and started to slide on THE wet asphalt. Then, he hit A parked car. He also crashed into two people walking on THE sidewalk. At 3:14, two officers arrived. Greg said he was not hurt badly, but he had A broken arm. They took him to THE hospital for AN examination.
Because my sweet and sour pork was cold, I put it in the microwave for a couple of minutes.
After six years in elementary school, still does not know how to play soccer.
John liked his new smartphone, but he didn't like the small screen.
The customers weren't buying the grapes, so the shopkeeper lowered the price.
The history of the iPod begins in 2001. Jon Rubinstein worked at Apple. He developed new products. He was in Japan looking for new products. He was in Japan looking for new technology. While touring a Toshiba factory, he saw a small hard drive. It was 4.5 cm long. The people at Toshiba said it was new, but they didn't know what to do with it.
Fill the 5-liter bucket. Pour water from the 5-liter bucket and fill the 3-liter bucket. Empty the 3-liter bucket. Pour the 2 liters from the 5-liter bucket into the 3-liter bucket. Now the 5-liter bucket is empty.
Fill the 5-liter bucket. Pour this water into the 3-liter bucket until full. You now have 4 liters of water in the 5-liter bucket.
Divide marbles into 3 groups – A, B, C. Each group has 3 marbles. Place A on one tray and B on the other. If the scale doesn't balance, you have found the group with a heavy marble. If the scales balance, group C has the heavy marble. Remove marbles.
From the heavy group, place one marble on each tray. If the scales balance, the extra marble is heavy. If scales don't balance, you have found the heavy marble.


text:
@interface UIFont (Prociono)

+ (instancetype)procionoRegularFontOfSize:(CGFloat)size;

@end

redpajama_set_name: RedPajamaGithub


Ohioans, we are in VERY REAL DANGER of losing an entire generation to addiction.
Ohio remains in the midst of an ongoing war against the opioid epidemic. Erick Fredrickson, a recovery coach, life coach, and former addict, is offering his experience as a way to help others cross the bridge into recovery. The following article originates from his blog, erickfredrickson.net.
The deadly effects of the "opioid crisis" is not new news in Ohio. For years running Ohio has been one of the worst states in the country for the number of overdose deaths per year. Currently, Ohio averages around 14 deaths a day from a drug overdose.
But the ramifications and ripple effects are slowly pulling much of a generation into the mess of addiction and the chaos it unleashes. For years I've been working with people that struggle with addiction and helping them live the life they are truly desiring, but more help is needed.
Imagine a child living in the same dirty diapers for days while home alone or even worse having to watch their parents do drugs in front of them, and all the insanity that that life entails. Imagine a child not eating while their parents spend the money on drug addiction. All the neglect, abuse, and unbelievably terrible things that come with that are literally ravaging thousands of defenseless children.
A common theme amongst the people I've worked with that struggle with addiction is the fact that they were exposed to addiction in their immediate family.
Ohioans, we can no longer pawn this horrific plague of drug addiction off on the police, courts, and treatment centers. It's all hands on deck. We must involve ourselves in the mess and fill the cracks that people are falling through into the abyss of death with love and hope.
The law and professionals are doing their best, the missing piece is the average citizen that lives next door to the families and individuals that have been devastated by the scars of addiction.


text:
NYCN urges FG to include minister of youth and sports in COVID 19 PTF
Posted on April 5, 2020 April 13, 2020 Author metro-adminComment(0) 447 Views
The National Youth Council of Nigeria (NYCN) has called on the Federal government to include the Minister of Youth and Sports Development, Mr Sunday Dare in the Presidential Task Force on Covid-19 to ensure coordinated outreach in terms of awareness and welfare.
President of NYCN, Mr Solomon Adodo, made the appeal in a statement he issued in Abuja on Sunday.
EditPro | Lagos Metropolitan newspaper Sunday Dare
Adodo said: "the Nigerian youths are already dying slowly under the biting pangs of hunger and this must be stopped through proper concerted efforts.
"There should be an immediate and direct release of relief materials and funds to all the 37 State Chapter Chairmen of the National Youth Council of Nigeria for onward distribution to the Nigerian youths.
"Nigerian youths should be fully and properly integrated into the activities of the Presidential Task Force on Covid-19 for faster and efficient results.
"It is importance that we alert the President that the Nigerian youths are the hardest hit by the stay at home directive which triggered a tactical shutdown of the economy.
"These youths were never captured amongst the poorest of the poor on account of the fact that they had means of eking a living for themselves howbeit on a subsistence day-to-day basis.
"This is no longer feasible and their lives are now in dire danger; this pressed situation might throw up unforeseen vicissitudes of grave implications if nothing is done to avert same," Adodo said.
The NYCN president who commended the government for the proactive measures initiated to contain the spread of the dreaded disease, called on the youths to remain law abiding and shun all forms of crimes as government looks into their demands.
He also applauded government's initiative to reach out to poor households amongst other worthy emergency measures to ease the adverse effects of the pandemic on the country.
"We call on all Nigerian youths to cooperate with the government at all levels and comply with constituted authority in our collective national interest.
"Government deserves our support now more than ever if we must win this war against COVID-19." (NAN)
100 business proposals to receive N100m grants from Osinbajo's friends
Sports Minister urges Nigerian athletes to keep hope alive in spite of coronavirus

redpajama_set_name: RedPajamaCommonCrawl


text:
Three new SingStar games coming to PS2, so let's get country!
Sony has announced 3 new versions of SingStar for the PS2, which include …
Andrew Webster - Sep 9, 2008 4:04 pm UTC
Despite the promise of downloadable tracks for the PlayStation 3 version of SingStar, Sony continues to roll out new discs full of songs for the PS2. This time around they have announced SingStar Country, Legends and Pop Volume 2.
The Country version will feature the likes Keith Urban, Johnny Cash, and Willie Nelson, while Legends will include everyone from John Lennon to Marvin Gaye to Black Sabbath. The second Pop edition meanwhile, will include more contemporary acts like Fallout Boy and Maroon 5. Head over to IGN for the complete track listings.
It's a bit disappointing for SingStar fans that these songs are only going to be available on the PS2, instead of also coming to the PS3 as downloadable content. Though the recently announced disc-swapping feature is better then nothing, actual DLC would be much preferred. It's hard to fault the business sense, however, as the PS2 has a much higher installed based than its big brother, and it's not like these titles need much horsepower to run.


Norman Bacon, the man, and his art are a mixture of spontaneity and control, perfection and imperfection. Each piece he has created is unique, an artistic expression bringing the material, fire, water, and air together. Norman started his a career in ceramics at the University of Wisconsin-Madison in the late sixties. His work with the raku process began in 1979. A marriage of material and enchantment was reborn. The Raku process consists of rapid firing and cooling, which transform clay objects into natural works of art. The resulting piece has colors and textures that cannot be duplicated.


text:
Buy, Sell, or Hold: Acme Packet
Shares tumbled on weak guidance; here's what investors need to know
Tamara Walsh
(TMFSocialME)
Published: Jul 9, 2012 at 12:00AM
I've been an analytical writer for The Motley Fool since 2011. I cover the sectors of Consumer Goods, Technology, and Industrials. Connect with me on Twitter using the handle, @TamaraRutter -- I'd love to hear from you! Follow @tamararutter
Shares of Acme Packet (Nasdaq: APKT) slid 16% last week, after the company issued weaker-than-expected guidance for its second quarter. With the stock trading at its lowest level in two years around $15 a pop, it could be an opportunity for investors to pick up shares at a discount. However, knowing when to buy, sell, or hold a stock can mean the difference between making money and losing it in the market. Let's take a closer look at Acme Packet and evaluate whether investors should buy, sell, or hold this networking equipment maker.
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Final answer?
Concerns over enterprise spending have put a lot of pressure on Acme. Still, from where I sit it doesn't seem management has a handle on the challenges it faces, which is why I think the stock is a sell today. Shares are down nearly 80% so far this year, and unfortunately, with growth in SBC slowing, the risks outweigh the benefits.
If you'd rather invest in a winning tech stock, I encourage you to read this free report from The Motley Fool: "The Next Trillion-Dollar Revolution." In it you will discover the next big opportunity for tech investors. Click here to get your copy of the free report while it's still available. Or if you'd like a better understanding of one of the biggest names in tech today, you can also grab a copy of our new premium research report on Apple , which you can access by just clicking here.


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I love pink, bling and everything else other girls like, but deep down I'm an absolute geek – enjoy my time on my computer, programming stuff, playing computer games (FPS, RTS, MMORPG, etc), watching animes (Naruto, Bleach, Lovely Complex, Kaichou Wa Maid-sama, Fairy Tail, etc), playing pool/billiard with the guys, etc.
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That experience taught me many things, especially the lesson on trust. After exiting my first somewhat failed business attempt not long after we started, I ventured into other businesses (e.g. pub) and eventually landed in the gaming space.
At 16-ish, I started a gaming network (powered by vBulletin); co-locating servers at data centres, then sponsored public game servers and rented private ones (mainly Counter-Strike) to local clans (and my own). Not too long after, I merged with another network, and together we ran the "Singapore Ragnarok Online". By (about) 18, I became the first vice president of probably the first registered gaming society in Singapore; and involved in organising/coordinating the World Cyber Games.
Through this gaming venture, I made a terrific friend, Neo Wee Teck (my tech guru), co-founder of EN Technologies (since 2001), and we've since been working together (somewhat).
Later years, more current times, I am the founder of PRINCESSA (fashion-lifestyle), Royal Privileged (branding) and SAUCEink (media), co-founder of SAUCE Networks (tech) and FESO Asia (entrepreneurship), director of marketing at iPaymy for Business (fintech), producer at EdSabLove, and pet of my business partners and friends. You can also check out my LinkedIn.
Public relations planning and management.
Digital and social media marketing.
Online, I'm better known as Princessa, an alter ego, online persona, pen name or however you want to call it, of the original version, which is myself, Sabrina Wang. The persona showcases a selective version of my actuality – a little more "ditsy", carefree, happy go lucky… and a little spoilt.
I started "blogging" (when the term "blogging" didn't even exist and "social media" was fiction) in the 1990s, a mix of personal and web development over a 56k modem. Days when Angelfire and Lycos were striving.
A lot has changed since, so did my supposed "branding".
When "blogging" popularised, I started adopting the nick "Princess Sabrina" (every girl wants to be a princess), and eventually shortened it to Princessa.
As the Princessa name caught on and the number of subscribers increased, it went from an online/blogging persona to somewhat a fashion brand – like Rachel Zoe, Marc Jacobs, etc – and PRINCESSA was born.
PRINCESSA is my (Sabrina Wang) fashion personality; featuring my fashion style and must-have wardrobe essentials. Every piece of apparel, bag, accessory, etc. is personally handpicked by me and can be found in my very own wardrobe.
Similar to the main brand, the PRINCESSA fashion label is where glamour meets wearable fashion. Fashion pieces that speak a lot about who wears it, PRINCESSA has masterfully crafted its taste in bringing fashion-forward items to every Fashionista.
Regardless if you are a Queen Bee, a sweet and demure Fashionista or a bold, fun-loving modern-day city gal, you will find something at PRINCESSA that fits you.
Established initially to create pieces that will make any woman feel like she's royalty, the demand for PRINCESSA clothing escalated. Now, at PRINCESSA, we provide a wider variety of top-notch modern fashion pieces at an affordable price range to better cater to the rising needs.
PRINCESSA is the brainchild of the renowned fashion-lifestyle blogger/influencer and entrepreneur Sabrina Wang. Her love for everything fabulous and glamorous has then sprouted to the idea, "Why not share my vision to the fashion world?" Since then, Sabrina has created fashion pieces touted as 'glamorous yet extremely wearable'. She has also shared to the fashion community her love for everything pink.
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Equinor Exits Ireland Following $434M Corrib Sale
by Bojan Lepic
Rigzone Staff
Equinor has stopped having a business presence in Ireland following a decision to sell its non-operated equity position in the Corrib offshore gas project to Vermillion Energy.
Norway's Equinor has stopped having a business presence in Ireland following a decision to sell its non-operated equity position in the Corrib offshore gas project to Vermillion Energy.
Equinor said it reached the agreement with Vermillion for the sale of its Corrib non-operated interest on consideration of $434 million, before closing adjustment, with an effective date set at January 1, 2022.
The deal, subject to approval by partners, government, and regulatory bodies, is expected to close during the second half of 2022.
As part of the transaction, Equinor and Vermilion have agreed to hedge approximately 70 percent of the production for 2022 and 2023 and have also agreed on a contingent payment that will be paid on a portion of the revenue if European gas prices exceed a given floor level.
The deal is subject to approval by partners, government, and regulatory bodies. The sale of Corrib means that Equinor will no longer have an active business presence in Ireland, after previously deciding to withdraw from an offshore wind project in the country.
The Corrib field started production in 2015 and is located 50 miles off Ireland's northwest coast in water depths of almost 1,150 feet. The equity gas volumes to Equinor for 2021 are estimated at around 58 million cubic feet per day.
The transaction is organized through a share sale of Equinor Energy Ireland Limited, a company 100 percent owned by Equinor. The Norwegian major owns 36.5 percent of the Corrib project, alongside Vermilion as the operator with 20 percent, and Nephin Energy with 43.5 percent.
"The Corrib field has been an important non-operated project for Equinor for several years. We have taken the decision to sell the asset to focus our portfolio, in line with our strategy, to capture value from the current strong market and to free up capital that we can re-invest elsewhere," Arne Gürtner, senior vice president responsible for the United Kingdom & Ireland, said.
To contact the author, email bojan.lepic@rigzone.com
Bojan Lepic
Editor | Rigzone
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