I've found out that one problem with my use of NDSolveValue is that using DirichletCondition might not be appropriate; I thought I'd checked that it worked in Cartesian coordinates but I must have mixed up data files exported from Mathematica because after checking again it looks like those conditions lead to errors even in the Cartesian case. So I've switched (back) to using another type of initial conditions. With these the Cartesian case works and I can also add an extra component to solve for without problems (the equations aren't coupled in Cartesian coordinates though so maybe there would still be an issue if they were coupled). The following shows what works in Cartesian coordinates:
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
eqz = D[Az[t, x], t, t] - D[Az[t, x], x, x] == 0
eqy = D[Ay[t, x], t, t] - D[Ay[t, x], x, x] == 0
omega := 0.2
Azic = Az[0, x] == - Sin[omega x]
dtAzic = Derivative[1, 0][Az][0, x] == -omega Cos[omega x]
Azicx = Az[t, 101] == - Sin[omega (101 + t)]
Ayic = Ay[0, x] == Cos[omega x]
dtAyic = Derivative[1, 0][Ay][0, x] == -omega Sin[omega x]
Ayicx = Ay[t, 101] == Cos[omega (101 + t)]
{Aysol, Azsol} =
NDSolveValue[{eqy, eqz, Ayic, dtAyic, Ayicx, Azic, dtAzic,
Azicx}, {Ay, Az}, {t, 0, 200}, {x, -101, 101}]
But doing the same in spherical coordinates leads to Mathematica computing until it finally just gives up, aborting the computation (seems to at least sometimes give the error that it runs out of memory). I even managed to reduce the problem to one just for A^r:
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
eqr = -r r D[Ar[t, r, phi], t, t] + r r D[Ar[t, r, phi], r, r] +
D[Ar[t, r, phi], phi, phi] + 4 r D[Ar[t, r, phi], r] +
2 Ar[t, r, phi] == 0
omega := 0.1
Aric = Ar[0, r, phi] == Sin[phi] Cos[omega (r Cos[phi])]
dtAric = Derivative[1, 0, 0][Ar][0, r,
phi] == -omega Sin[phi] Sin[omega (r Cos[phi])]
Arb = Ar[t, 150, phi] ==
Sin[phi] Cos[omega (150 Cos[phi] + t)]
Arperiodic = Ar[t, r, 0] == Ar[t, r, 2 Pi]
{Arsol} =
NDSolveValue[{eqr, Aric, dtAric, Arb, Arperiodic}, {Ar}, {t, 0,
100}, {r, 0.001, 150}, {phi, 0, 2 Pi} ]
I find it quite odd that this equation should be too difficult for Mathematica? I'm currently trying to see if using a smaller interval for t and r in the solution region will help.