Copyright (C) 2020 Andreas Kloeckner
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
NB: This note book uses the Maxima-Jupyter kernel to interface with Maxima, an open-source computer algebra system. I have found that the Docker image is a fairly convenient way of using this kernel.
Some Maxima version info for reproducibility:
build_info();
Load some packages:
kill(all);
load("itensor");
assert(condition):=if not condition then error("Assertion violated") else true$
norm_2_squared(v):=v.v;
crossfunc(f):=makelist(
sum(sum(
levi_civita([i,j,k])*f(j,k),
j,1,3),k,1,3),i,1,3)$
crossprod(a,b):=crossfunc(lambda([j,k], a[j]*b[k]));
REDEFINITION-WITH-DEFMACRO: redefining MAXIMA::IFNOT in DEFMACRO REDEFINITION-WITH-DEFMACRO: redefining MAXIMA::M+OR*OR^P in DEFMACRO REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDUMMY-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDUMMY in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ISPROD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DERAT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::PLUSI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::MINUSI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::COVI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DERI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::NAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COVI-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COVI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CONTI-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CONTI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$DERI-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$DERI in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$NAME-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$NAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$DEFCON in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::GETCON in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::RPOBJ in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IMETRIC-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IMETRIC in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::MYSUBST0 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::COV in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTR in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DIFFCOV in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ICHR1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ICHR2 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ICURVATURE-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ICURVATURE in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::COVSUBST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONSUBST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DERSUBST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DIFFOP in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COVDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::COVDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::COVDIFFTIMES in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::VECDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::LIEDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LIEDIFF-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LIEDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REDIFF-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$EVUNDIFF-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$EVUNDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$UNDIFF-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$UNDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::PUTINONES in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LORENTZ_GAUGE in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::LORENTZ in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::LESS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CONTRACT-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CONTRACT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT4A in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT5 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::HEAD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::FIRSTINTERSECT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT2 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SETDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT3 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT4 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$RENORM-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$RENORM in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::NEGLIST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ABNORM in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SUBSTLIST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::REMOVENOTIN in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::REMOVENOTINM in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACTINSIDE in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONTRACT1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$KDELTA-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$KDELS-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$KDELS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DELTA in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ISHOW-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ISHOW in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ISHOW in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SPLICE in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SPLICE1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SPLICE2 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DERIV in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CHAINRULE1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DIFFEXPT1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFMAP in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFTIMES in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFEXPT1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFEXPT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFINT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFINT1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFF%DERIV in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDERIV in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFNCEXPT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFSUMPROD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFFGRAD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDIFF-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::IDIFF in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::DIFFRPOBJ in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LC0-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LC0 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LEVI_CIVITA-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LC_L in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LC_U in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::NONUMBER in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::REMOVEINDEX in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::INDICES in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDICES-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDICES in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SAMELISTS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$FLUSH in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$FLUSHD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::FLUSH in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$FLUSHND-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$FLUSHND in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$RENAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::RENAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::REORDER in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ITENSOR-CLEANUP in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CLEANUP1 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ITENSOR-SORT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REMCOMPS-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REMCOMPS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDEXED_TENSOR-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDEXED_TENSOR in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::ALLFIXED in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::TENSOREVAL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COMPONENTS-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COMPONENTS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::SELECT in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$ENTERTENSOR in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CHECKINDEX in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::MEMBERL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CONSMLIST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDICES2-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$INDICES2 in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CHANGENAME-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$CHANGENAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::CHANGENAME in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$COORD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REMCOORD-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$REMCOORD in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LISTOFTENS-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$LISTOFTENS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::LISTOFTENS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::NUMLIST in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$SHOWCOMPS-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$SHOWCOMPS in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$HODGE-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$HODGE in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::REMSYM in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDIM-IMPL in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::$IDIM in DEFUN REDEFINITION-WITH-DEFUN: redefining MAXIMA::I-$DEPENDENCIES in DEFUN
These function simplify expressions coming from the (symbolic) simultaneous diagonalization, by letting Maxima realize that $n \cdot n =1$.
/* ------------------------------------------------------------------------- */
/* Simplification for expressions stemming from hyperbolic systems */
/* ------------------------------------------------------------------------- */
hypsimp(x):=ratsimp(ratsubst(1,n.n,x))$
fullhypsimp(x):=hypsimp(
ratsubst(
last(n)^2,
1-sum(n[i]^2,i,1,length(n)-1),
x)
)$
/* ------------------------------------------------------------------------- */
/* diagonalize a given hyperbolic operator A */
/* ------------------------------------------------------------------------- */
hypdiagonalize(A):=block([evA, V, invV,D],
evA:hypsimp(apply(append, eigenvectors(A)[2])),
V:transpose(apply(matrix, evA)),
invV:hypsimp(invert(V)),
assert(hypsimp(V.invV)=ident(length(A))),
D:hypsimp(invV.A.V),
[V, D, invV])$
/* redefine this to change dimensionality: */
n:[nx,ny];
dims:length(n);
assume(c>0);
if dims = 1 then n:[1];
Define the flux jacobian for the first-order form of the wave-equation:
$$ \begin{align*} \partial_t u &= \nabla \cdot \boldsymbol v\\ \partial_t \boldsymbol v &= \nabla u \end{align*} $$projected onto a line with normal n
:
esymmatrix(n, v, i,j):=ematrix(n,n,v,i,j)+ematrix(n,n,v,j,i);
wave_A:sum(n[i]*esymmatrix(dims+1, -c, 1+i,1),i,1,dims);
Find the eigenvalues of the flux Jacobian:
[wave_V, wave_D, wave_invV]:hypdiagonalize(wave_A);
This function sets up a system of Rankine-Hugoniot conditions across the flux fan and solves for the stagnation state:
/* ------------------------------------------------------------------------- */
/* compute upwind flux for a given operator with eigenvalues evs, sorted
* in ascending order.
* Sign assumptions for all variables occuring in evs must be in place.
*/
/* ------------------------------------------------------------------------- */
hyp_upwind_flux(evs, D):=block([evvars, Dp, Dm, n, midstates, states, unknowns],
evvars:listofvars(evs),
add_evvars_suffix(suffix, x):=subst(makelist(v=concat(''v, suffix), v, evvars), x),
evsm:add_evvars_suffix(m, evs),
evsp:add_evvars_suffix(p, evs),
Dm:add_evvars_suffix(m, D),
Dp:add_evvars_suffix(p, D),
midstates:makelist(makelist(concat(s,state,i), i, 1, length(D)),
state, 1, length(evs)-1),
states:append(
[makelist(concat(sm, i), i, 1, length(D))],
midstates,
[makelist(concat(sp,i), i, 1, length(D))]),
unknowns:apply(append, midstates),
result:if member(0, evs) then
block([biasedD, veceqns, eqns, soln],
biasedD:makelist(
if evs[i] = 0 then [Dp,Dm]
else if evs[i] > 0 then [Dp,Dp]
else [Dm,Dm],
i, 1, length(evs)),
veceqns:apply(append, makelist(
-(if evs[i] > 0 then evsp[i] else evsm[i]) *(states[i+1]-states[i])
+(biasedD[i][1].states[i+1]-biasedD[i][2].states[i]),
i,1,length(evs))),
eqns:makelist(veceqns[i,1], i, 1, length(veceqns)),
soln:solve(eqns, unknowns),
assert(length(soln)=1),
for i: 1 thru length(evs) do
if evs[i] = 0 then return(Dp.subst(soln[1], midstates[i]))
)
else
block([straddle_idx, Dstates, veceqns, eqns, soln],
straddle_idx:for i: 1 thru length(evs)-1 do
if (evs[i] < 0) and (evs[i+1] > 0) then return(i),
flux:makelist(concat(flux,i),i,1,length(D)),
unknowns:append(unknowns, flux),
Dstates:append(
[Dm.first(states)],
makelist(
if i = straddle_idx then flux
else if evs[i] > 0 then Dp.midstates[i]
else Dm.midstates[i],
i, 1, length(midstates)),
[Dp.last(states)]),
veceqns:apply(append, makelist(
-(if evs[i] > 0 then evsp[i] else evsm[i]) *(states[i+1]-states[i])
+(Dstates[i+1]-Dstates[i]),
i,1,length(evs))),
eqns:makelist(veceqns[i,1], i, 1, length(veceqns)),
print(covect(eqns)),
soln:solve(eqns, unknowns),
assert(length(soln)=1),
subst(soln[1], flux)
),
subst(
append(
makelist(concat(sm, i)=sm[i,1], i, 1, length(D)),
makelist(concat(sp, i)=sp[i,1], i, 1, length(D))
),
result)
)$
Find an expression for the flux in characteristic variables.
Note the p
and m
suffixes for the $+$ and $-$ sides of the interface.
wave_eigenvalues:makelist(wave_D[i,i], i, 1, length(wave_D));
if member(0, wave_eigenvalues) then
wave_sflux:hyp_upwind_flux([-c,0,c], wave_D)
else
wave_sflux:hyp_upwind_flux([-c,c], wave_D);
solve: dependent equations eliminated: (1 6 8)
Convert back to conserved variables:
wave_wflux:ratsimp(wave_V.ev(wave_sflux, [sm=wave_sminw,sp=wave_spinw]));
First, set up some parameter assumptions:
assume(c>0);
assume(mu>0);
assume(epsilon>0);
assume(epsinv>0);
assume(muinv>0);
Some helper functions for matrix creation:
/* A hyperbolic system matrix resulting from a curl */
curlmat(coord):=genmatrix(
lambda ([i,j], levi_civita([coord,j,i])),
3,3)$
vstack:append$
hstack(a,b):=transpose(append(transpose(a),transpose(b)))$
blockmat(a11,a12,a21,a22):=vstack(hstack(a11,a12),hstack(a21,a22))$
n:[nx,ny,nz];
Next, write down the flux Jacobian on a line with normal n
:
max_submat(i):=blockmat(
zeromatrix(3,3),
-epsinv*curlmat(i), /* epsinv = 1/epsilon */
muinv*curlmat(i), /* muinv = 1/mu */
zeromatrix(3,3)
)$
max_Asimp:sum(n[i]*max_submat(i),i,1,3);
max_A:subst([epsinv=1/epsilon,muinv=1/mu], max_Asimp);
Next, diagonalize to obtain the transformation to/from characteristic variables:
max_invsubst(x):=subst([epsinv=1/epsilon, muinv=1/mu], x)$
[max_V, max_D, max_invV]:max_invsubst(hypdiagonalize(max_Asimp));
Now find the flux in characteristic variables:
max_Dinc:subst([1/(sqrt(epsilon)*sqrt(mu))=c], max_D);
max_sflux:hyp_upwind_flux([-c,0,c], max_Dinc);
/* FIXME: max_V should not depend on epsilon and mu, but it does
For now, make cp and cm equal. */
max_sflux:subst(
[cp=1/(sqrt(epsilon)*sqrt(mu)), cm=1/(sqrt(epsilon)*sqrt(mu))],
max_sflux);
solve: dependent equations eliminated: (1 2 11 16 15 12)
And in conserved variables:
max_Em:makelist(Em[i],i,1,3)$
max_Ep:makelist(Ep[i],i,1,3)$
max_Hm:makelist(Hm[i],i,1,3)$
max_Hp:makelist(Hp[i],i,1,3)$
max_wm:vstack(max_Em,max_Hm)$
max_wp:vstack(max_Ep,max_Hp)$
max_sminw:hypsimp(max_invV.max_wm)$
max_spinw:hypsimp(max_invV.max_wp)$
max_wflux:fullhypsimp(max_V.ev(max_sflux, [sm=max_sminw,sp=max_spinw]));
max_stronglocalpart:max_A.max_wm;
max_strongwflux:max_stronglocalpart-max_wflux;
Check against value from the literature:
max_Z:sqrt(mu/epsilon)$
max_Y:sqrt(epsilon/mu)$
max_knownstrongwflux:ratsimp(vstack(
-1/(2*epsilon)
*(crossprod(n,(max_Hm-max_Hp)-1/max_Z*crossprod(n,max_Em-max_Ep))),
1/(2*mu)
*(crossprod(n,(max_Em-max_Ep)+1/max_Y*crossprod(n,max_Hm-max_Hp)))
));
assert(norm_2_squared(hypsimp(
(max_strongwflux)
-max_knownstrongwflux))=0);