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so - called numerical relativity ( computational simulations in general relativity ) is a promising research field having implications for ongoing astrophysical observations such as gravitational wave astronomy @xcite .
many simulations of binary compact objects have revealed that mathematically equivalent sets of evolution equations show different numerical stability in the free - evolution scheme .
there are many approaches to re - formulate the einstein equations for obtaining a longterm stable and accurate numerical evolution ( e.g. see references in @xcite ) . in a series of our works
, we have proposed the construction of a system that has its constraint surface as an attractor . by applying eigenvalue analysis of constraint propagation equations
, we showed that there _ is _ a constraint - violating mode in the standard arnowitt - deser - misner ( adm ) evolution system @xcite when it is applied to a single non - rotating black - hole space - time@xcite .
we also found that such a constraint - violating mode can be compensated for if we adjust the evolution equations with a particular modification using constraint terms like the one proposed by detweiler @xcite .
our predictions are borne out in simple numerical experiments using the maxwell , ashtekar , and adm systems @xcite .
there are also several numerical experiments to confirm our predictions are effective@xcite .
however we have not yet obtained definite guidelines for specifying the above adjusting terms and their multipliers .    in this article
, we show the essential steps in analyzing constraint amplification factors ( defined in  [ seccafdef ] ) . in  [ sec3 ] , we show that degeneracy of eigenvalues sometimes leads constraint evolution to diverge .
this observation suggests the importance of checking the diagonalizability of characteristic matrices , and gives further insights for constructing an asymptotically constrained system .
we begin by reviewing our proposal for an  adjusted system " .
suppose we have a dynamical system of variables @xmath0 , which has evolution equations , @xmath1 and the ( first class ) constraints , @xmath2 note that we do not require ( [ ueq ] ) to form a first - order hyperbolic form .
we propose to investigate the evolution equation of @xmath3 ( constraint propagation ) , @xmath4 for evaluating violation features of constraints .
the character of constraint propagation , ( [ ceq ] ) , will vary when we modify the original evolution equations .
suppose we modify ( adjust ) ( [ ueq ] ) using constraints @xmath5 then ( [ ceq ] ) will also be modified as @xmath6 therefore , finding a proper adjustment @xmath7 is a quite important problem .
hyperbolicity analysis may be a way to evaluate constraint propagation , ( [ ceq ] ) and ( [ ceqadj ] ) @xcite
. however , this requires ( [ ceq ] ) to be a first - order system which is easy to be broken .
( see e.g. detweiler - type adjustment @xcite in the adm formulation @xcite ) .
furthermore hyperbolicity analysis only concerns the principal part of the equation , that may fail to analyze the detail evaluation of evolution .
alternatively , we have proceeded an eigenvalue analysis of the whole rhs in ( [ ceq ] ) and ( [ ceqadj ] ) after a suitable homogenization , which may compensate for the above difficulties of hyperbolicity analysis .
we propose to transform the constraint propagation equation , ( [ ceq ] ) and ( [ ceqadj ] ) , into fourier modes , @xmath8 then to analyze the eigenvalues , say @xmath9 , of the coefficient matrix , @xmath10 .
we call @xmath9 and @xmath10 the constraint amplification factors ( cafs ) and constraint propagation matrix ( cp matrix ) , respectively .
so far we have proposed the following heuristic conjectures @xcite :    * if the caf has a _ negative real - part _ ( the constraints are necessarily diminished ) , then we see more stable evolution than a system which has a positive caf . *
if the caf has a _ non - zero imaginary - part _ ( the constraints are propagating away ) , then we see more stable evolution than a system which has a zero caf .
we observe that this eigenvalue analysis requires the fixing of a particular background space - time , since the cafs depend on the dynamical variables , @xmath11 .
the cafs indicate the evolution of constraint violations ( definitely its fourier modes ) .
it is natural to assume that a divergence of constraint norm is related to the numerical blow - ups .
therefore we classify the fundamental evolution property of constraint propagation equation ( [ ceqf ] ) as follows :    * _ asymptotically constrained : _ violation of constraints decays ( converges to zero ) . * _ asymptotically bounded : _ violation of constraints is bounded at a certain value . * _ diverge : _ at least one constraint will diverge .    note that ( c1 ) @xmath12 ( c2 ) .
we will derive the necessary and sufficient conditions for ( c1 ) and ( c2 ) in the next section .
hereafter , we consider a set of evolution equations , @xmath13 where @xmath14 is a complex - valued vector , @xmath15 is a @xmath16 complex - valued matrix , and @xmath17 is assumed to have finite - valued initial data @xmath18 .    without loss of generality , the cp matrix @xmath19 can be assumed to be a jordan normal form , since within complex - valued operations all the matrices can be converted to this form .
suppose that @xmath19 has @xmath20 different eigenvalues @xmath21 , where @xmath22 .
let the multiplicity of @xmath23 as @xmath24 , where @xmath25 .
@xmath19 can be expressed as @xmath26 where the cell size of @xmath27 is @xmath28 .
the jordan matrix @xmath27 is then expressed using a jordan block @xmath29 , @xmath30 note that @xmath29 is @xmath31 , @xmath32 , @xmath33 and @xmath34 .
the minimum polynomial of @xmath19 is written as @xmath35 if @xmath27 is diagonal ( i.e. @xmath36 ) , then @xmath37 for that @xmath38 . if @xmath19 is diagonalizable ( i.e. @xmath36 for @xmath39 ) , then @xmath37 for all @xmath38 .
we then have the following statement .
the solution of @xmath40 can be expressed formally as @xmath41 [ propa ]    a proof is available by mathematical induction .
suppose that @xmath42 is @xmath43 which is the maximal size @xmath27 . by direct calculation
, we have that @xmath44 yields ( [ kino ] ) with @xmath45-polynomial of degree @xmath46 . then we see that ( [ kino ] ) is satisfied in general .    from this proposition , the highest power @xmath47 in @xmath48polynomial in ( [ kino ] )
is bounded by @xmath49 .
the matrix @xmath27 in ( [ cp2 ] ) can be directly extended to the full cp matrix , @xmath19 , in ( [ cp ] ) .
therefore the highest power @xmath50 in all constraints is bounded by @xmath51      the following propositions [ prop1 ] and [ prop2 ] give us the next theorem .
asymptotically constrained evolution ( violation of constraints converges to zero ) is obtained if and only if all the real parts of the cafs are negative .
[ theoremc1 ]    all the real part of cafs are negative @xmath52 asymptotically constrained evolution .
[ prop1 ]    proof ) we use the expression ( [ kino ] ) .
if @xmath53 for @xmath39 , then @xmath54 will converge to zero at @xmath55 no matter what the @xmath48polynomial terms are .
@xmath56    asymptotically constrained evolution @xmath52 all the real parts of the cafs are negative .
[ prop2 ]    proof ) we show the contrapositive .
suppose there exists an eigenvalue @xmath57 of which the real - part is non - negative .
then we get @xmath58 of which the solution is @xmath59 .
@xmath60 does not converge to zero .
@xmath56      the following propositions [ prop3 ] and [ prop4 ] give us the next theorem .     asymptotically bounded evolution ( all the constraints are bounded at a certain value ) is obtained if and only if all the real parts of cafs are not positive and @xmath27 is diagonal when @xmath61 .
[ theoremc2 ]    * corollary * _ asymptotically bounded evolution is obtained if the real parts of cafs are not positive and the cp matrix @xmath62 is diagonalizable .
_    all the real parts of cafs are not positive and @xmath27 is diagonal when @xmath61 @xmath52 asymptotically bounded evolution .
[ prop3 ]    proof ) we use the expression ( [ kino ] ) .
when @xmath63 , @xmath64 will converge to zero no matter what the @xmath48polynomial terms are .
when @xmath61 , we see @xmath37 from the assumption of diagonality of @xmath27 .
so we see the @xmath48polynomial terms are constant and @xmath65 is bounded . @xmath56    asymptotically bounded evolution @xmath52 all the real parts of the cafs are not positive and @xmath27 is diagonal when @xmath66 .
[ prop4 ]    proof ) we show the contrapositive .
if there exists an eigenvalue of which the real - part is positive , then constraints will diverge no matter what the @xmath48polynomial terms are
. therefore we try to show that constraints will diverge when all the real - parts of eigenvalues are non - positive , and there exists @xmath23 such that @xmath66 and its jordan matrix @xmath27 is not diagonal .
since jordan matrix @xmath27 is not diagonal , we see the power of @xmath48polynomial @xmath67 is greater than 1 in the expression ( [ kino ] ) .
thus we have that ( [ kino ] ) will diverge in @xmath68 . @xmath56
two theorems will give us a guideline to analyze a constraint - violating mode of the system .
the result supports our previous heuristic conjecture ( a ) , but also suggests an ill - behaving case when cafs are degenerated and its real - part is zero , when the associated jordan matrix is not diagonal .
this indicates the importance of checking the diagonalizability of constraint propagation matrix @xmath19 .    along the line of our evaluation of constraint propagation equations ( [ cp ] )
, we propose a practical procedure for this classification in figure [ fig : flowchart ] .
we think that this diagram will provide systematic predictions for obtaining a robust evolution system in any constrained dynamics .
( 85,75 ) ( -5,00 )    the present classification is only on the fixed background spacetime and only for @xmath69 .
it is still not clear at what value the constraints are bounded if a limiting value exists .
thus further modifications are underway .
we are also applying the present classification scheme to various adjusted systems of the einstein equations ( adjusted adm , and further modified versions ) , together with numerical experiments .
we hope to report on them in the near future .
the current constraint analysis only concentrates to the evolution equations and does not include the effect of the boundary treatments . since the eigenvalues are evaluated locally , it will be possible to include the effect of numerical boundary conditions if they are expressed apparently in a part of the evolution equations .
this is also the one direction to proceed our future research .
meanwhile , we would like to remark that one of our proposed adjustments in @xcite contributes to enforce the computational ability of the black - hole excision boundary treatment @xcite .    by extending the notion of  norm " or  compactness " of constraint violations
, it might be interesting to define a new measure which monitors a  distance " between the constraint surface and an evolution sector in constraint dynamics .
hs thanks the caltech visitors program for the numerical simulation of gravitational wave sources for their hospitality , where a portion of this work was completed .
hs is supported by the special postdoctoral researchers program at riken .
this work was supported partially by the grant - in - aid for scientific research fund of japan society of the promotion of science , no .
14740179 .
h. shinkai and g. yoneda ,  reformulating the einstein equations for stable numerical simulations " , in _ progress in astronomy and astrophysics _ ( nova science publ ) to be published .
the manuscript is available upon request .
in order to obtain stable and accurate general relativistic simulations , re - formulations of the einstein equations are necessary . in a series of our works 
 , we have proposed using eigenvalue analysis of constraint propagation equations for evaluating violation behavior of constraints . in this article 
 , we classify asymptotical behaviors of constraint - violation into three types ( asymptotically constrained , asymptotically bounded , and diverge ) , and give their necessary and sufficient conditions . 
 we find that degeneracy of eigenvalues sometimes leads constraint evolution to diverge ( even if its real - part is not positive ) , and conclude that it is quite useful to check the diagonalizability of constraint propagation matrices . 
 the discussion is general and can be applied to any numerical treatments of constrained dynamics .     
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