Structure‐preserving numerical methods for Fokker–Planck equations

A common way to numerically solve Fokker–Planck equations is the Chang–Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, th...

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Bibliographic Details
Published inProceedings in applied mathematics and mechanics Vol. 24; no. 4
Main Authors Bartel, Hanna, Lampert, Joshua, Ranocha, Hendrik
Format Journal Article
LanguageEnglish
Published 01.12.2024
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Summary:A common way to numerically solve Fokker–Planck equations is the Chang–Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang–Cooper method with unconditionally positive Patankar‐type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher‐order accurate. We describe the combined approach, analyse it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature.
ISSN:1617-7061
1617-7061
DOI:10.1002/pamm.202400007