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

TL;DR

This paper addresses the challenge of preserving the maximum principle in high-order finite-volume schemes for hyperbolic conservation laws by directly comparing a priori slope limiters with a posteriori, MOOD-style limiters. It implements Zhang & Shu's MPP a priori limiter and a MOOD-based a posteriori approach with a MUSCL fallback, testing them on linear advection in 1D and 2D across multiple polynomial degrees $p$ and Runge–Kutta time integrators. Key findings show that in 1D the a priori limiter strictly preserves the maximum principle across all $p$, while in 2D the a priori approach can incur large violations or diffusion with cheap flux reconstructions; the a posteriori schemes deliver higher long-time solution quality and cost efficiency, especially with transverse flux reconstruction, albeit with bound violations that can be mitigated by blending. The study highlights a fundamental trade-off between strict bound preservation and numerical diffusion/cost, and demonstrates substantial performance gains from GPU acceleration. All mathematical notation in this summary is presented within $...$ delimiters where appropriate, e.g., $p$, $M$, $m$, $C$, and $t^{n}$.

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. ...

Figures (14)

• Figure 1: The edges of the stability regions of forward Euler, SSPRK2, SSPRK3, RK4, and RK6 are shown in the top left panel. The subsequent five panels show each Runge-Kutta method overlayed with the eigenvalue tracks of $D=\mathcal{L}_p$ corresponding to finite volume upwinding and a CFL factor $C=1$. The tracks are shown for $p$ from 0 to 7, linearly shaded from purple to yellow.
• Figure 2: The value of $\phi$ used for convex blending of corrected fluxes, shown across a region comprising seven cells, with one troubled cell highlighted in red.
• Figure 3: Cell nodal values used to compute cell fluxes (red) and the a priori slope limiter $\theta$ (blue) for polynomial degree $p=5$.
• Figure 4: The value of $\phi$ used for the convex blending of corrected fluxes, shown across a region comprising 49 cells with troubled cells highlighted in red.
• Figure 5: Snapshots of the numerical solution to the advection of the composite profile at $t=1$. Results are shown for the aPrioriMPP and aPosterioriB schemes and for polynomial degrees $p=2$ (dark blue), $p=3$ (light blue), and $p=7$ (yellow). The aPrioriMPP schemes use SSPRK3 for all results shown while the aPosterioriB schemes use SSPRK3 for $p=2$ and RK4 for $p>2$. The numerical solution of second-order MUSCL-Hancock is shown in dashed grey for reference. All results shown have a resolution of $N=256$ cells. Maximum principle violations are not observed in the aPrioriMPP and MUSCL-Hancock solutions while they are observed for the aPosterioriB solutions.