Large time behaviors of upwind schemes by jump processes
For numerical analysts, it provides a new probabilistic perspective and rigorous convergence results for upwind schemes in unbounded domains, but the results are incremental as they extend known exponential convergence to new settings.
The paper revisits upwind schemes for linear conservation laws using jump processes, proving uniform exponential convergence to steady states in the whole space (without boundary) and on torus, with O(h) error under ℓ¹ norm for strongly confining Fokker-Planck equations.
We revisit the traditional upwind schemes for linear conservation laws in the viewpoint of jump processes, allowing studying upwind schemes using probabilistic tools. In particular, for Fokker-Planck equations on $\mathbb{R}$, in the case of weak confinement, we show that the solution of upwind scheme converges to a stationary solution. 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 upwind schemes, our result is in the whole space without boundary. We also establish similar results on torus for which the stationary solution of the scheme does not have detailed balance. This work shows an interesting connection between standard numerical methods and time continuous Markov chains, and could motivate better understanding of numerical analysis for conservation laws.