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Mathematics of Computation

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

Authors: Lei Li and Jian-Guo Liu
Journal: Math. Comp. 89 (2020), 2283-2320
MSC (2010): Primary 65M12, 65M75; Secondary 65C05
Published electronically: February 18, 2020
MathSciNet review: 4109567
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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.

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Additional Information

Lei Li
Affiliation: School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

Jian-Guo Liu
Affiliation: Department of Mathematics and Department of Physics, Duke University, Durham, North Carolina 27708

Keywords: Upwind scheme, discrete Poincar\'e inequality, unbounded domain, detailed balance, semigroup
Received by editor(s): July 23, 2018
Received by editor(s) in revised form: September 9, 2019, and November 25, 2019
Published electronically: February 18, 2020
Additional Notes: The first author was sponsored in part by NSFC 11901389, 11971314, and Shanghai Sailing Program 19YF1421300.
The second author was supported in part by KI-Net NSF RNMS11-07444 and NSF DMS-1812573.
Article copyright: © Copyright 2020 American Mathematical Society