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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "read(\"d:/gem31.txt\"):" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%doGEM~v.031.28.6~~<<DARMSTADT>>~Copyright(C)~Alexei~F.
~Cheviakov,~2004-2009G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "
#conservation laws of the 3D static MHD equations\n#Six scalar, plus o
ne infinite family (in the current multiplier ansatz)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "ind:=x,y,z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ind
G6%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "gem_d
ecl_vars(indeps=[ind], deps=[B1(ind),B2(ind),B3(ind),P(ind)]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#B1G6%%\"xG%\"yG%\"zG\"\"\"%9will~
now~be~displayed~asGF*F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#B2G
6%%\"xG%\"yG%\"zG\"\"\"%9will~now~be~displayed~asGF*F%F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*(-%#B3G6%%\"xG%\"yG%\"zG\"\"\"%9will~now~be~dis
played~asGF*F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"PG6%%\"xG%\"
yG%\"zG\"\"\"%9will~now~be~displayed~asGF*F%F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%AIndependent~variables:~[x,~y,~z]G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%EDependent~variables:~[B1,~B2,~B3,~P]G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%3Free~functions:~[]G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%3Free~constants:~[]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%XDefinition~of~variables~succesful;~~~GeM~is~initializedG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gP:=grad(P(ind),[ind]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gPGK%'vectorG6#7%&%\"PG6#%\"xG&F*6#
%\"yG&F*6#%\"zGQ(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "B_vect:=[B1(ind),B2(ind),B3(ind)];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'B_vectG7%%#B1G%#B2G%#B3G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "divB:=diverge(B_vect,[ind]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%divBG,(&%#B1G6#%\"xG\"\"\"&%#B2G6#%\"yGF*&%#B3G6#%\"
zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "curlB:=curl(B_vect,
[ind]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&curlBGK%'vectorG6#7%,&&%
#B3G6#%\"yG\"\"\"&%#B2G6#%\"zG!\"\",&&%#B1GF1F.&F+6#%\"xGF3,&&F0F8F.&F
6F,F3Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cuBxB:
=crossprod(curlB,B_vect);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&cuBxBG
K%'vectorG6#7%,&*&,&&%#B1G6#%\"zG\"\"\"&%#B3G6#%\"xG!\"\"F0F2F0F0*&,&&
%#B2GF3F0&F-6#%\"yGF5F0F9F0F5,&*&F7F0F-F0F0*&,&&F2F;F0&F9F.F5F0F2F0F5,
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "gem_de
cl_eqs(\n[\ncuBxB[1]=gP[1],\ncuBxB[2]=gP[2],\ncuBxB[3]=gP[3],\ndivB=0
\n],\nsolve_for=[\ndiff(P(ind),x),\ndiff(P(ind),y),\ndiff(P(ind),z),\n
diff(B1(ind),x)\n]\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~1
~~(Order=1):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%#B3G\"\"\"%$B1z
GF'F'*&F&F'%$B3xGF'!\"\"*&%#B2GF'%$B2xGF'F+*&F-F'%$B1yGF'F'%#PxGF+\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~2~~(Order=1):G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%#B1G\"\"\"%$B2xGF'F'*&F&F'%$B1yG
F'!\"\"*&%#B3GF'%$B3yGF'F+*&F-F'%$B2zGF'F'%#PyGF+\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%7Equation~3~~(Order=1):G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,,*&%#B2G\"\"\"%$B3yGF'F'*&F&F'%$B2zGF'!\"\"*&%#B1GF'%
$B1zGF'F+*&F-F'%$B3xGF'F'%#PzGF+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%7Equation~4~~(Order=1):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%$B
1xG\"\"\"%$B2yGF&%$B3zGF&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%UNo
~differential~consequences~required~on~this~stage.G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%?Equation~definition~successfulG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 72 "det_eqs:=gem_conslaw_det_eqs(\n[\nind,  B1(
ind),B2(ind),B3(ind),P(ind)\n]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Y
Generating~determining~equtions~for~conservation~laws...G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CL_multipliers:=gem_conslaw_multipl
iers();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/CL_multipliersG7&-%(Lamb
da1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG-%(Lambda2GF(-%(Lambda3GF(-%(
Lambda4GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "simplified_eq
s:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%/simplified_eqsGK%&TABLEG6#7$/%*dimensionG%)in
finityG/%'SolvedG7@/&%(Lambda4G6$%\"yGF3\"\"!/&F16$F3%\"zGF4/&F16$%\"P
GF3F4/&%(Lambda2G6$F8F8F4/&F1F@F4/&F16$F<F8F4/&F16$F<F<,$*&&%(Lambda3G
6#F<\"\"\"%#B3G!\"\"FP/&%(Lambda1G6#%\"xGF4/&F?FT*&,&&F16#F3FP*&&F?6#F
8FNFOFNFPFN%#B1GFP/&FLFT*&,&&F1FhnFP*&FgnFN%#B2GFNFNFNFinFP/&F1FT*&,&*
&F`oFNFZFNFP*&FOFNF^oFNFPFNFinFP/&FSFen*&,&FZFNFfnFNFNFinFP/&F?FenF4/&
FLFen,$FgnFP/&FSFhn*&,&F^oFNF_oFPFNFinFP/&FLFhnF4/&FS6#Fin,$&F1FMFP/&F
?FhpF4/&FLFhpF4/&F1Fhp,&-FS6)FUF3F8FinF`oFOF<FN*&FjpFNFinFNFN/&FS6#F`o
F4/&F?FgqFip/&FLFgqF4/&F1Fgq,&-F?FcqFN*&FjpFNF`oFNFN/&FS6#FOF4/&F?FcrF
4/&FLFcrFip/&F1Fcr,&*&FjpFNFOFNFN-FLFcqFN/&FSFM*(FKFNFinFNFOFP/&F?FM*(
FKFNF`oFNFOFPQ(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "multipliers_sol:=pdsolve(simplified_eqs[Solved]);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%0multipliers_solG<&/-%(Lambda1G6)%\"xG%\"yG%\"zG
%#B1G%#B2G%#B3G%\"PG,**&F+\"\"\"%$_C2GF3F3*&F,F3%$_C3GF3F3%$_C4GF3*&&%
%_F10G6#F0F3F-F3!\"\"/-%(Lambda2GF),**&F,F3%$_C1GF3F<*&F*F3F4F3F<%$_C5
GF3*&F9F3F.F3F</-%(Lambda3GF),**&FBF3F+F3F3%$_C6GF3*&F*F3F6F3F<*&F9F3F
/F3F</-%(Lambda4GF),*-F:F;F3*&,(F2F3F5F3F7F3F3F-F3F3*&,(FAF<FCF<FDF3F3
F.F3F3*&,(FJF3FKF3FLF<F3F/F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#Infinite set of
 cons. laws, depending on an arbitrary function." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
166 "#Since the multipliers involve arbitrary function, we can only us
e Anco's scaling method. \n#Luckily, the PDE system has an obvious sca
ling symmetry X=Bi D_Bi + 2P D_P" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "gem_ge
t_CL_fluxes(multipliers_sol, method=\"Scaling\", symmetry=\{eta_B1=B1,
eta_B2=B2,eta_B3=B3,eta_P=2*P\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
%'MethodGQ(Scaling6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%^^lMultiplie
rs:~[Lambda1(x,y,z,B1,B2,B3,P)~=~y*_C2+z*_C3+_C4-diff(_F10(P),P)*B1,~L
ambda2(x,y,z,B1,B2,B3,P)~=~-z*_C1-x*_C2+_C5-diff(_F10(P),P)*B2,~Lambda
3(x,y,z,B1,B2,B3,P)~=~_C1*y+_C6-x*_C3-diff(_F10(P),P)*B3,~Lambda4(x,y,
z,B1,B2,B3,P)~=~_F10(P)+(y*_C2+z*_C3+_C4)*B1+(-z*_C1-x*_C2+_C5)*B2+(_C
1*y+_C6-x*_C3)*B3]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q-Symmetries:~6
\"K%'vectorG6#7&%#B1G%#B2G%#B3G,$*&\"\"#\"\"\"%\"PGF/F/Q(pprint0F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%7----------------------G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%coVerifying~div(fluxes)=0~using~solutions~and~
differential~consequences...G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Ind
eed,~div(fluxes)=0!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7------------
----------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ho-----~Full~Fluxes~(m
aybe~involving~arbitrary~const.~and/or~functions):~------G" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG,.*&)%#B2G\"\"#\"\"\",**&%\"yGF*%$
_C2GF*F**&%\"zGF*%$_C3GF*F*%$_C4GF**&&%%_F10G6#%\"PGF*%#B1GF*!\"\"F*F9
*&)%#B3GF)F*F+F*F9*(F)F*F7F*F+F*F9*(F(F*,**&F0F*%$_C1GF*F9*&%\"xGF*F.F
*F9%$_C5GF**&F4F*F(F*F9F*F8F*F**(F<F*,**&FAF*F-F*F*%$_C6GF**&FCF*F1F*F
9*&F4F*F<F*F9F*F8F*F**&F8F*,*-F5F6F**&,(F,F*F/F*F2F*F*F8F*F**&,(F@F9FB
F9FDF*F*F(F*F**&,(FHF*FIF*FJF9F*F<F*F*F*F*" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/%'Flux_yG,.*(%#B1G\"\"\",**&%\"yGF(%$_C2GF(F(*&%\"zGF(
%$_C3GF(F(%$_C4GF(*&&%%_F10G6#%\"PGF(F'F(!\"\"F(%#B2GF(F(*&)F'\"\"#F(,
**&F.F(%$_C1GF(F6*&%\"xGF(F,F(F6%$_C5GF(*&F2F(F7F(F6F(F6*&)%#B3GF:F(F;
F(F6*(F:F(F5F(F;F(F6*(FDF(,**&F=F(F+F(F(%$_C6GF(*&F?F(F/F(F6*&F2F(FDF(
F6F(F7F(F(*&F7F(,*-F3F4F(*&,(F*F(F-F(F0F(F(F'F(F(*&,(F<F6F>F6F@F(F(F7F
(F(*&,(FHF(FIF(FJF6F(FDF(F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%'
Flux_zG,.*(%#B1G\"\"\",**&%\"yGF(%$_C2GF(F(*&%\"zGF(%$_C3GF(F(%$_C4GF(
*&&%%_F10G6#%\"PGF(F'F(!\"\"F(%#B3GF(F(*(%#B2GF(,**&F.F(%$_C1GF(F6*&%
\"xGF(F,F(F6%$_C5GF(*&F2F(F9F(F6F(F7F(F(*&)F'\"\"#F(,**&F<F(F+F(F(%$_C
6GF(*&F>F(F/F(F6*&F2F(F7F(F6F(F6*&)F9FCF(FDF(F6*(FCF(F5F(FDF(F6*&F7F(,
*-F3F4F(*&,(F*F(F-F(F0F(F(F'F(F(*&,(F;F6F=F6F?F(F(F9F(F(*&,(FEF(FFF(FG
F6F(F7F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@------------------
-------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SCases~separated~w.
r.t.~free~constants~/~functions:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
@-------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
D------------Case~C1~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6&/-%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG\"\"!/-%(Lambda2GF
&,$F)!\"\"/-%(Lambda3GF&F(/-%(Lambda4GF&,&*&F)\"\"\"F+F<F3*&F(F<F,F<F<
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%'Flux_xG,&**\"\"#\"\"\"%#B2GF(%
\"zGF(%#B1GF(!\"\"**F'F(%#B3GF(%\"yGF(F+F(F(/%'Flux_yG,,*&)F+F'F(F*F(F
(*&)F.F'F(F*F(F(*(F'F(%\"PGF(F*F(F(**F'F(F.F(F/F(F)F(F(*&F*F()F)F'F(F,
/%'Flux_zG,,**F'F(F)F(F*F(F.F(F,*&F4F(F/F(F,*&F;F(F/F(F,*(F'F(F8F(F/F(
F,*&F/F(F6F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------Case~C
2~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%(Lambda1G6)%
\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PGF(/-%(Lambda2GF&,$F'!\"\"/-%(Lambda3
GF&\"\"!/-%(Lambda4GF&,&*&F(\"\"\"F*F<F<*&F'F<F+F<F2" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%/%'Flux_xG,,*&)%#B2G\"\"#\"\"\"%\"yGF*!\"\"*&F+F*)%#
B3GF)F*F,*(F)F*%\"PGF*F+F*F,**F)F*F(F*%\"xGF*%#B1GF*F,*&)F4F)F*F+F*F*/
%'Flux_yG,,**F)F*F4F*F+F*F(F*F**&F6F*F3F*F**&F.F*F3F*F**(F)F*F1F*F3F*F
**&F3F*F'F*F,/%'Flux_zG,&**F)F*F/F*F+F*F4F*F***F)F*F(F*F3F*F/F*F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------Case~C3~---------------G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G
%#B2G%#B3G%\"PGF)/-%(Lambda2GF&\"\"!/-%(Lambda3GF&,$F'!\"\"/-%(Lambda4
GF&,&*&F)\"\"\"F*F<F<*&F'F<F,F<F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/
%'Flux_xG,,*&%\"zG\"\"\")%#B2G\"\"#F(!\"\"*&)%#B3GF+F(F'F(F,*(F+F(%\"P
GF(F'F(F,**F+F(F/F(%\"xGF(%#B1GF(F,*&)F4F+F(F'F(F(/%'Flux_yG,&**F+F(F*
F(F'F(F4F(F(**F+F(F*F(F3F(F/F(F,/%'Flux_zG,,**F+F(F4F(F'F(F/F(F(*&F6F(
F3F(F(*&F3F(F)F(F(*(F+F(F1F(F3F(F(*&F.F(F3F(F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%D------------Case~C4~---------------G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6&/-%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG
\"\"\"/-%(Lambda2GF&\"\"!/-%(Lambda3GF&F2/-%(Lambda4GF&F*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%/%'Flux_xG,**$)%#B2G\"\"#\"\"\"!\"\"*$)%#B3GF)F
*F+*&F)F*%\"PGF*F+*$)%#B1GF)F*F*/%'Flux_yG,$*(F)F*F3F*F(F*F*/%'Flux_zG
,$*(F)F*F3F*F.F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D------------Ca
se~C5~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%(Lambda1G
6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG\"\"!/-%(Lambda2GF&\"\"\"/-%(Lamb
da3GF&F./-%(Lambda4GF&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%'Flux_xG
,$*(\"\"#\"\"\"%#B1GF(%#B2GF(F(/%'Flux_yG,**$)F)F'F(!\"\"*$)%#B3GF'F(F
0*&F'F(%\"PGF(F0*$)F*F'F(F(/%'Flux_zG,$*(F'F(F*F(F3F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%D------------Case~C6~---------------G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&/-%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G
%\"PG\"\"!/-%(Lambda2GF&F./-%(Lambda3GF&\"\"\"/-%(Lambda4GF&F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%/%'Flux_xG,$*(\"\"#\"\"\"%#B1GF(%#B3GF
(F(/%'Flux_yG,$*(F'F(%#B2GF(F*F(F(/%'Flux_zG,**$)F)F'F(!\"\"*$)F/F'F(F
5*&F'F(%\"PGF(F5*$)F*F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%E-----
-------Case~F10~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-
%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG,$*&&%\"FG6#F-\"\"\"F*F
3!\"\"/-%(Lambda2GF&,$*&F0F3F+F3F4/-%(Lambda3GF&,$*&F0F3F,F3F4/-%(Lamb
da4GF&-F1F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%'Flux_xG*&%#B1G\"\"\"
,&*(\"\"#F'%\"PGF'&%\"FG6#F+F'F'-F-F.F'F'/%'Flux_yG*&%#B2GF'F(F'/%'Flu
x_zG*&%#B3GF'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@---------------
----------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%T~=========~Cons
.~law~flux~splitting~DONE~=========~G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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