Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws
Jianfang Lu
School of Mathematics, South China University of Technology, Canton, Guangdong, China
Search for more papers by this authorCorresponding Author
Yan Jiang
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
Correspondence
Yan Jiang, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China.
Email: [email protected]
Search for more papers by this authorChi-Wang Shu
Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA
Search for more papers by this authorMengping Zhang
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
Search for more papers by this authorJianfang Lu
School of Mathematics, South China University of Technology, Canton, Guangdong, China
Search for more papers by this authorCorresponding Author
Yan Jiang
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
Correspondence
Yan Jiang, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China.
Email: [email protected]
Search for more papers by this authorChi-Wang Shu
Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA
Search for more papers by this authorMengping Zhang
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
Search for more papers by this authorAbstract
In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.
Open Research
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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