A Positivity-Preserving Relaxation Algorithm

TL;DR

This paper introduces a relaxation algorithm that ensures positivity and conservation in numerical methods, applicable to ordinary and partial differential equations, validated through theoretical analysis and numerical experiments.

Contribution

It combines Patankar-type methods with relaxation procedures to produce unconditionally positive, conservative approximations with new solvability conditions.

Findings

The algorithm guarantees positivity and conservation in numerical schemes.

Sufficient conditions for solvability of the coupled equations are established.

Numerical experiments validate the theoretical properties of the method.

Abstract

We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative approximations. To that end, we adapt the relaxation algorithm to enforce positivity by using either ideas from the dense output framework when a linear invariant must be preserved, or simply a geometric mean if the only constraint is positivity preservation. The latter merely requires the solution of a scalar nonlinear equation while former results in a coupled linear-nonlinear system of equations. We present sufficient conditions for the solvability of the respective equations. Several applications in the context of ordinary and partial differential equations are presented, and the theoretical findings are validated numerically.