2024/04/28 by Jonathan Palafoutas, Palafoutas, Jonathan, David A Velasco Romero +3
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2404.18037
openalex publication_date 2024/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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: a priori and a posteriori limiters. The former revises the high-order solution based only on data at the current time tn, while the latter involves computing a candidate solution at tn+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 a priori and a posteriori limiters for finite volume solvers at arbitrarily high order. We select the maximum principle preserving scheme presented in \citezhang2011maximum, zhang2010maximum as our a priori limited scheme. For a posteriori limiting, we adopt the methodology presented in \citeclain2011high and search for so-called 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. ...