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Analysis of Arbitrary High Order Spectral Volume Method for Hyperbolic Conservation Laws Over Rectangular Meshes
Waixiang Cao, Zhimin Zhang and Qingsong Zou

J. Comp. Math. DOI: 10.4208/jcm.2504-m2024-0201

Publication Date : 2025-09-05

  • Abstract

This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the $L^2$ norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.

  • Copyright

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