Numerical solution of the hyperbolic equation

Question

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.

Answer 1

Comment: I think you want D instead of Derivative. Also == instead of =. And you probably want the functions defined with patterns z_ etc. But there are errors that you'll have to address. (Or perhaps someone else.)

ClearAll[φ, η, r, u];
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];

pde = D[u[t, ρ, z], {t, 2}] == 
   D[u[t, ρ, z], {ρ, 2}] + (1/ρ)*D[u[t, ρ, z], {ρ, 1}] + 
    D[u[t, ρ, z], {z, 2}] - ((φ[z]^4)/4)*
      (D[(η[z] + u[t, ρ, z] - 1)/φ[z], {ρ, 1}] D[(η[z] + u[t, ρ, z] + 1)/φ[z], {ρ, 1}] + 
       D[(η[z] + u[t, ρ, z] - 1)/φ[z], {z, 1}] D[(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);
bc = {u[0, ρ, z] == 0,
   u[t, 0, z] == 1,
   u[t, ρ, -6] == u[t, ρ, 6]};

NDSolve[{pde, bc}, 
 u[t, ρ, z], {t, 0, 10}, {ρ, 0, 6}, {z, -6, 6}, 
 Method -> {"MethodOfLines", 
   "SpatialDiscretization" -> {"TensorProductGrid", 
     "DifferenceOrder" -> "Pseudospectral"}}]

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>

I'll leave this up for a while, but I'll probably delete it since it is not an answer. :( Unless I forget. Maybe someone else can use it.

Update

q and k have to be given numeric values for NDSolve to work. In NDSolve, ρ starts at 0 and there's a 1/ρ in the PDE, which will give you a 1/0 error (when the NDSolve finally works). There may be other such singularities to deal with, but this one was obvious.

Here NDSolve goes to work but gives the 1/0 error. The trick to get it to work is to give a complete IVP for t. The highest order derivative is two, so initial values for u and its t derivative have to be given. (I do not know if this is how you should fix your problem.)

q = 2; k = 1/10; (* random values)
bc = {u[0, ρ, z] == 0, 
   Derivative[1, 0, 0][u][0, ρ, z] == 1,
   u[t, 0, z] == 1,
   u[t, ρ, -6] == u[t, ρ, 6]};
NDSolve[{pde, bc},
 u[t, ρ, z],
 {t, 0, 10}, {ρ, 0, 6}, {z, -6, 6}, 
 Method -> {"MethodOfLines", 
   "SpatialDiscretization" -> {"TensorProductGrid", 
     "DifferenceOrder" -> "Pseudospectral"}}]

At this point, I believe I am stuck. The problem, I believe, is now a mathematical/scientific one of what the appropriate boundary and initial conditions are. It is also possible that there is an error (typo) in the PDE.

If you need to vary q and k as parameters, consider using ParametricNDSolve. But the basic NDSolve problem should be fixed first.

Sorry I can't be more help.