Numerical methods for highly-oscillatory problems

We constructed numerical time integrators for nonlinear Dirac equations and the one-dimensional Klein–Gordon system.

Numerical methods for highly-oscillatory nonlinear Dirac equations

The code that can be downloaded below analyses the approximation error of the numerical methods for the semi-nonrelativistic limit system of the nonlinear Dirac equation presented in On numerical methods for the semi-nonrelativistic limit system of the nonlinear Dirac equation by Tobias Jahnke and Michael Kirn. Two main functions are provided. The first one analyses the error in dependency of the timestep size, whereas the second one analyses the error in dependency of computation time. The code was written in Matlab by Michael Kirn.

Requirements

The code requires Matlab version 2021a or later.

Download

Download software package (zip, 29kB)

The one-dimensional Klein–Gordon system

The first part of the code is an implementation of the slowly-varying envelope equation (SVEA) and its extension for the one-dimensional Klein–Gordon system, written in Matlab by Julian Baumstark. For further details on the SVEA and its extension have a look at the doctoral thesis by Julian Baumstark.

The second part of the code contains two numerical time integrators for the SVEA without any step-size restrictions. For further information about the derivation and error analysis of those numerical integrators have a look at the respective publication.

Requirements

Details about the requirements can be found in the README file and the documentation at https://publikationen.bibliothek.kit.edu/1000149721.

Download

https://publikationen.bibliothek.kit.edu/1000149721 (Chapter_4 and Chapter_5).

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.