Conservative-dissipative approximation schemes for a generalized Kramers equation

TL;DR

This paper introduces three novel discrete variational schemes that effectively model the conservative and dissipative dynamics of a generalized Kramers equation, with proven convergence to the true solution.

Contribution

The paper presents new variational schemes inspired by large deviations theory, specifically designed for the generalized Kramers equation, and establishes their convergence.

Findings

All three schemes converge to the solution of the generalized Kramers equation.

The schemes incorporate large deviations rate functionals into their design.

Two schemes are single-step minimizations, and one combines streaming with minimization.

Abstract

We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes while the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.