Numerical methods for nonlinear PDEs in large domains

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Discussion Overview

The discussion centers on numerical methods for solving nonlinear partial differential equations (PDEs) over large domains, specifically in the context of z-propagated nonlinear Schrödinger equations related to nonlinear optics. Participants explore various approaches to improve computational efficiency and stability in simulations involving long-range dynamics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes their use of a pseudo-spectral method with adaptive step size control via a Runge-Kutta-Fehlberg technique, noting issues with computation times over long propagation distances.
  • Another participant questions the nature of z-propagated PDEs and suggests deriving the CFL criterion to understand the relationship between mesh size and time step for stability.
  • Further inquiries are made regarding the order of the Runge-Kutta-Fehlberg method and the characteristics of the nonlinearities in the problem, emphasizing the importance of understanding whether the problem is convection or diffusion dominant.
  • A participant mentions that the problem involves simulating intense pulse propagation through nonlinear materials and provides details about the time step and spatial grid size.
  • Suggestions are made to consider stiff integration methods if the resulting ODEs from discretization are stiff, and to potentially transform spatial coordinates for better handling of large distances.

Areas of Agreement / Disagreement

Participants express varying opinions on the best approaches to take for solving the problem, with no consensus reached on a single method or solution. Multiple competing views and suggestions remain present throughout the discussion.

Contextual Notes

Participants note the dependence of suggestions on specific problem characteristics, such as the nature of the nonlinearities and the dominant physical processes, indicating that the discussion is context-sensitive and lacks a one-size-fits-all solution.

Who May Find This Useful

This discussion may be useful for researchers and practitioners in the fields of nonlinear optics, numerical analysis, and computational physics, particularly those dealing with nonlinear PDEs and their numerical solutions over large domains.

paddy.
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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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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?
 
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.
 
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.
 
Thanks, i'll look into some your suggestions.
 

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