A Structure Preserving Scheme for the Kolmogorov-Fokker-Planck Equation

TL;DR

This paper introduces a numerical scheme that preserves the long-term behavior of solutions to the Kolmogorov-Fokker-Planck equation by transforming it into a form suitable for structure-preserving discretization.

Contribution

It proposes a novel self-similar transformation approach to develop structure-preserving schemes for the Kolmogorov-Fokker-Planck equation.

Findings

The scheme accurately captures long-time dynamics.

Operator splitting enhances computational efficiency.

Numerical results validate the scheme's effectiveness.

Abstract

In this paper we introduce a numerical scheme which preserves the long time behavior of solutions to the Kolmogorov equation. The method presented is based on a self-similar change of variables technique to transform the Kolmogorov equation into a new form, such that the problem of designing structure preserving schemes, for the original equation, amounts to building a standard scheme for the transformed equation. We also present an analysis for the operator splitting technique for the self-similar method and numerical results for the described scheme.