Numerical methods for nonlinear PDEs in large domains

In summary, the conversation discusses the issue of solving z-propagated nonlinear PDEs numerically over large domains. The current approach of using a pseudo-spectral method with adaptive step size control is effective for short propagation lengths but has limitations for longer distances. The conversation also delves into potential solutions such as using a CFL criterion for mesh size and time step, trying smaller time steps, and using a higher order RKF method. The specific problem being discussed is nonlinear optics and the nonlinearities are strong in the material. Additional suggestions are given, including using a stiff integration package and transforming spatial coordinates for large distances.
  • #1
paddy.
3
0
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 don't 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 don't 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!
 
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  • #2
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
bigfooted said:
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?

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 Schrödinger 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
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
Thanks, i'll look into some your suggestions.
 

FAQ: Numerical methods for nonlinear PDEs in large domains

What are numerical methods for solving nonlinear PDEs?

Numerical methods for solving nonlinear PDEs are techniques used to approximate solutions to partial differential equations that involve nonlinear terms and are difficult to solve analytically. These methods involve discretizing the PDE and using iterative algorithms to find numerical solutions.

Why are numerical methods necessary for solving PDEs in large domains?

In large domains, analytical solutions to PDEs are often impossible to find due to the complexity of the equations. Therefore, numerical methods are necessary to approximate solutions and make the problem more manageable.

What are some common numerical methods for nonlinear PDEs?

Some common numerical methods for solving nonlinear PDEs include finite difference methods, finite element methods, and spectral methods. These methods involve approximating the solution on a discrete grid and using iterative techniques to find the solution.

How do numerical methods handle nonlinear terms in PDEs?

Numerical methods handle nonlinear terms in PDEs by linearizing them through iterative processes. This involves making an initial guess for the solution and updating it using the nonlinear terms until a satisfactory solution is reached.

What challenges are associated with using numerical methods for nonlinear PDEs in large domains?

Some challenges associated with using numerical methods for nonlinear PDEs in large domains include the need for a fine grid resolution, which can be computationally expensive, and the potential for instability or convergence issues. It is also important to choose the appropriate numerical method for the specific problem at hand.

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