Large time behaviors of upwind schemes and $B$-schemes for Fokker-Planck equations on $\mathbb {R}$ by jump processes

Abstract:

We revisit some standard schemes, including upwind schemes and some $B$-schemes, for linear conservation laws from the viewpoint of jump processes, allowing the study of them using probabilistic tools. For Fokker-Planck equations on $\mathbb {R}$, in the case of weak confinement, we show that the numerical solutions converge to some stationary distributions. In the case of strong confinement, using a discrete Poincaré inequality, we prove that the $O(h)$ numeric error under $\ell ^1$ norm is uniform in time, and establish the uniform exponential convergence to the steady states. Compared with the traditional results of exponential convergence of these schemes, our result is in the whole space without boundary. We also establish similar results on the torus for which the stationary solution of the scheme does not have detailed balance. This work could motivate better understanding of numerical analysis for conservation laws, especially parabolic conservation laws, in unbounded domains.