I am trying to solve the following hyperbolic equation with given boundary conditions:
I choose as initial condition $u=1$, and evolve the above hyperbolic equation until reaching a stationary state with some predetermined tolerance.
Spatial derivatives are approximated by taking standard centered differences everywhere.
The Mathematica input is:
φ[z] = q*(1/z + (-1*q)/(-1*z));
η[z] := k* (1/z + (-1*q)/(-1*z));
r[ρ, z] := sqrt[ρ^2 + z^2];
NDSolve[ {Derivative[u[t, ρ, z], {t, 2}] ==
Derivative[u[t, ρ, z], {ρ, 2}] + (1/ρ)*
Derivative[u[t, ρ, z], {ρ, 1}] +
Derivative[
u[t, ρ, z], {z, 2}] - ((φ[z]^4)/4)* (
Derivative[(η[z] + u[t, ρ, z] - 1)/φ[
z], {ρ,
1}] Derivative[(η[z] + u[t, ρ, z] +
1)/φ[z], {ρ, 1}] +
Derivative[(η[z] + u[t, ρ, z] - 1)/φ[
z], {z,
1}] Derivative[(η[z] + u[t, ρ, z] +
1)/φ[z], {z, 1}]) *((η[z] +
u[t, ρ,
z])*((η[z] + u[t, ρ, z])^2 - (φ[z]^2)/
2) )^(-1), u[0, ρ, z] = 0, u[t, 6, z] = 1,
u[t, ρ, -6] = u[t, ρ, 6]},
u[t, ρ, z], {t, 0, 10}, {ρ, 0, 6}, {z, -6, 6},
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"DifferenceOrder" -> "Pseudospectral"}}]
which gives errors.
I would like to ask if you could help me to solve this problem.