Maximum principle preserving space and time flux limiting for Diagonally Implicit Runge-Kutta discretizations of scalar convection-diffusion equations

TL;DR

This paper introduces a high-order discretization framework for scalar convection-diffusion equations that ensures maximum principle preservation without step size restrictions, using a two-tiered limiting strategy.

Contribution

It develops a novel high-order, maximum-principle-preserving scheme with a two-step limiting process for scalar convection-diffusion equations, ensuring stability and accuracy.

Findings

Schemes achieve high-order accuracy in space and time.

The methods are stable and preserve the maximum principle without step size restrictions.

Numerical examples demonstrate the effectiveness of the proposed schemes.

Abstract

We provide a framework for high-order discretizations of nonlinear scalar convection-diffusion equations that satisfy a discrete maximum principle. The resulting schemes can have arbitrarily high order accuracy in time and space, and can be stable and maximum-principle-preserving (MPP) with no step size restriction. The schemes are based on a two-tiered limiting strategy, starting with a high-order limiter-based method that may have small oscillations or maximum-principle violations, followed by an additional limiting step that removes these violations while preserving high order accuracy. The desirable properties of the resulting schemes are demonstrated through several numerical examples.