An explicit Euler method for Sobolev vector fields with applications to the continuity equation on non cartesian grids

2024/02/06 by Tommaso Cortopassi, Cortopassi, Tommaso · 1 citation
Engineering · #35L65 #65L07 #65M15 #65M75 #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2402.04118

openalex publication_date 2024/02/06 · openalex created_date 2024/02/08 · openalex updated_date 2026/08/03

Abstract

We prove a novel stability estimate in L^∞ _t (L^p _x) between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of) explicit Euler method, and it is the crucial tool to prove approximation results for the solution of the continuity equation by using the representation of the solution as the push-forward via the regular Lagrangian flow of the initial datum. We approximate the solution in two ways, one probabilistic and one deterministic, using different approximations for both the flow and the initial datum. Such estimates for the solution of the continuity equation are derived on non Cartesian grids and without the need to assume a CFL condition.

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