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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.18037 |
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| _version_ | 1866914773585625088 |
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| author | Palafoutas, Jonathan Romero, David A. Velasco Teyssier, Romain |
| author_facet | Palafoutas, Jonathan Romero, David A. Velasco Teyssier, Romain |
| contents | 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.
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| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes Palafoutas, Jonathan Romero, David A. Velasco Teyssier, Romain Numerical Analysis 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. ... |
| title | Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2404.18037 |