Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

TL;DR

This paper develops and analyzes semi-Lagrangian schemes on unstructured grids for viscous transport and conservative equations with one-sided Lipschitz velocity fields, ensuring convergence through viscosity and duality solutions.

Contribution

It introduces a novel convergence analysis for semi-Lagrangian schemes applied to equations with measurable coefficients under one-sided Lipschitz conditions.

Findings

Schemes converge to the true solutions in various numerical tests.

Theoretical analysis confirms the schemes' stability and accuracy.

Numerical examples illustrate the effectiveness of the proposed methods.

Abstract

The aim of this work is to investigate semi-Lagrangian approximation schemes on unstructured grids for viscous transport and conservative equations with measurable coefficients that satisfy a one-sided Lipschitz condition. To establish the convergence of the schemes, we exploit the characterization of the solution for these equations expressed in terms of measurable time-dependent viscosity solution and, respectively, duality solution. We supplement our theoretical analysis with various numerical examples to illustrate the features of the schemes.