Higher Order Multidimensional Slope Limiters with Local Maximum Principles

TL;DR

This paper develops a flexible slope limiter framework for higher-order finite volume schemes on unstructured meshes, enabling adherence to local maximum principles with less restrictive limiting, thus maintaining higher accuracy.

Contribution

It introduces a new framework for slope limiting that achieves local maximum principles with reduced limiting in higher-order schemes on unstructured meshes.

Findings

Framework successfully applied to second-order schemes.

Framework extended to fourth-order schemes.

Maintains accuracy while enforcing maximum principles.

Abstract

Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations and equations of transport type. Limiting is required to either converge to the correct type of solution or to adhere to physically motivated local maximum principles. Less restrictive limiting procedures are required so as to not severely decrease the accuracy. In this paper, we develop an existing slope limiter framework, to achieve different local boundedness principles for higher-order schemes on unstructured meshes. Quadrature points contributing to numerical fluxes can be limited based on face defined maximum principles, and the resulting cell mean at the next timestep can satisfy a cell mean maximum principle but with less limiting. We demonstrate the practical application of the introduced framework to a second-order finite volume scheme as well as a fourth-order finite volume scheme, in the context of the advection equation.