The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws

TL;DR

This paper extends the Lax-Wendroff theorem to include Patankar-type methods for hyperbolic conservation laws, demonstrating their convergence to weak solutions under certain conditions, supported by numerical validation.

Contribution

It provides the first theoretical extension of the Lax-Wendroff theorem to nonlinear Patankar-type schemes with a total variation boundedness assumption.

Findings

The extended theorem confirms convergence of Patankar schemes to weak solutions.

Numerical simulations support the theoretical results.

Patankar schemes are unconditionally conservative and positivity-preserving.

Abstract

For hyperbolic conservation laws, the famous Lax-Wendroff theorem delivers sufficient conditions for the limit of a convergent numerical method to be a weak (entropy) solution. This theorem is a fundamental result, and many investigations have been done to verify its validity for finite difference, finite volume, and finite element schemes, using either explicit or implicit linear time-integration methods. Recently, the use of modified Patankar (MP) schemes as time-integration methods for the discretization of hyperbolic conservation laws has gained increasing interest. These schemes are unconditionally conservative and positivity-preserving and only require the solution of a linear system. However, MP schemes are by construction nonlinear, which is why the theoretical investigation of these schemes is more involved. We prove an extension of the Lax-Wendroff theorem for the class of MP methods. This is the first extension of the Lax--Wendroff theorem to nonlinear time integration methods with just an additional hypothesis on the total time variation boundedness of the numerical solutions. We provide some numerical simulations that validate the theoretical observations.