RNPL solution of Carlos Palenzuela et al's model equation discussed in MPI-AEI Numerical Relativity Seminar 2008-04-15

Equations of motion

w = w(t,x)
v = v(t,x)

   w_t = v_x

   v_t = w_x + (-v + c w) / epsilon

on 0 <= x <= 1, t >= 0, with initial conditions

   v(0,x) = v0(x)
   w(0,x) = (v0 + exp(-(x - 0.5)^2/0.01) * epsilon) / c

and boundary conditions

   v(t,0) = v(t,1) = 0.

The initial conditions are chosen so that for epsilon -> 0, the initial data is consistent with a pure left-moving solution, while for epsilon of O(1), the solution has both left and right moving parts.

Unless otherwise specified, c = 1 in the following.

Discretization

(Iterative) Crank-Nicholson time differencing, second order centred spatial differencing.

• RNPL source code [wvadv_rnpl]
• RNPL parameter file [id0]
  

Results

MPEG animation of v(t,x) for epsilon = 1, 10^-2, 10^-4, 10^-6, 10^-8, 10^-10 (top to bottom, curves have been offset vertically for visualization purposes).

• Courant factor dt/dx = 0.5 in all cases
• Crank-Nicholson tolerance: 10^-6
• Number of Crank-Nicholson iterations to converge to fixed tolerance generally increases roughly as log_10(1/epsilon)

Again, there is no evidence of instability in the Crank-Nicholson iteration, although for epsilon < 10^-12, there are
problems with convergence that I suspect are due to "round off" (i.e. finite precision arithmetic).

Additionally, convergence tests indicate O(h^2) convergence in all cases.

MPEG animation of v(t,x) for epsilon = 1, 10^-2, 10^-4, 10^-6, 10^-8, 10^-10 (top to bottom, curves have been offset vertically for visualization purposes) with number of iterations fixed to 2.  No apparent difference from variable-iteration results.

MPEG animation as above, but with c = 0.5.

Maintained by [email protected]. Supported by CIFAR, NSERC, CFI, BCKDF and UBC.