# Numerical methods for nonlinear PDEs in large domains

1. Mar 19, 2013

Hi all, first post :)

I have a system of z-propagated nonlinear PDEs that I solve numerically via a pseudo-spectral method which incorporates adaptive step size control using a Runge-Kutta-Fehlberg technique. This approach is fine over short propagation lengths but computation times dont scale well with propagation distance.

My question is are there any special approaches for solving PDEs over very large domains? If so can these be applied to NL equations? I dont require exact solutions just some idea of the long range dynamics.

Thanks for reading, more information can be supplied if what I have asked isn't clear. I also apologise if I just asked the mathematical equivalent of turning lead into gold, i'm an experimentalist who has accidentally become a theorist, so be kind!

2. Mar 20, 2013

### bigfooted

What are z-propagated PDE's? What kind of problem is this exactly?

Can you derive the CFL criterion for your problem or a simplified version of it? It gives you a relation between your mesh size and your time step, so something like dt/dx < C where C is PDE-dependent and discretization-dependent, involving PDE-dependent variables. So your time step has to be smaller than some constant times the mesh size to be stable at all. Have you tried using very small time steps?

What is the order of your RKF method? For time dependent problems involving e.g. turbulence, some people use 12th order schemes.

A lot of further suggestions depend on your specific problem. Is it convection or diffusion dominant? How strong are the nonlinearities? What kind of nonlinearities?

3. Mar 20, 2013

Thank you for replying. Its a nonlinear optics problem. I'm simulating the propagation of intense pulses through nonlinear materials with a system of nonlinear SchrÃ¶dinger equations.

The time step is of the order of femtoseconds and the total size of the spatial grid is currently several mm, but i wish to extend this considerably hence my question.

The RKF method is 5th order and the nonlinearities are fairly strong in my material.

4. Mar 21, 2013

### Staff: Mentor

Welcome to Physics Forums.

There are probably lots of things you can do. If the set of ODE you are using as a result of discretizing the spatial variables is stiff, the Runge Kutta is not a good choice. Try using a stiff integration package. If you are extending to large distances, and don't need very fine resolution on the solution at large distances, you can try transforming the spatial coordinates.

5. Mar 24, 2013