Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes

TL;DR

This paper compares a priori and a posteriori slope limiters in high-order finite volume schemes, evaluating their effectiveness in maintaining solution quality, maximum principle adherence, and computational efficiency for hyperbolic conservation laws.

Contribution

It introduces a methodology to compare these two limiting paradigms at arbitrarily high order for finite volume methods, including practical implementation details.

Findings

A posteriori limiters effectively identify troubled cells and improve solution quality.

A priori limiters better preserve the maximum principle with lower computational cost.

Trade-offs exist between solution accuracy, principle preservation, and computational expense.

Abstract

High-order finite volume and finite element methods offer impressive accuracy and cost efficiency when solving hyperbolic conservation laws with smooth solutions. However, if the solution contains discontinuities, these high-order methods can introduce unphysical oscillations and severe overshoots/undershoots. Slope limiters are an effective remedy, combating these oscillations by preserving monotonicity. Some limiters can even maintain a strict maximum principle in the numerical solution. They can be classified into one of two categories: \textit{a priori} and \textit{a posteriori} limiters. The former revises the high-order solution based only on data at the current time $t^n$, while the latter involves computing a candidate solution at $t^{n+1}$ and iteratively recomputing it until some conditions are satisfied. These two limiting paradigms are available for both finite volume and finite element methods. In this work, we develop a methodology to compare \textit{a priori} and \textit{a posteriori} limiters for finite volume solvers at arbitrarily high order. We select the maximum principle preserving scheme presented in \cite{zhang2011maximum, zhang2010maximum} as our \textit{a priori} limited scheme. For \textit{a posteriori} limiting, we adopt the methodology presented in \cite{clain2011high} and search for so-called \textit{troubled cells} in the candidate solution. We revise them with a robust MUSCL fallback scheme. The linear advection equation is solved in both one and two dimensions and we compare variations of these limited schemes based on their ability to maintain a maximum principle, solution quality over long time integration and computational cost. ...