An Oscillation-free Spectral Volume Method for Hyperbolic Conservation Laws

TL;DR

This paper introduces an oscillation-free spectral volume method for hyperbolic conservation laws, incorporating damping to suppress oscillations near discontinuities while maintaining accuracy and convergence.

Contribution

The paper proposes a novel OFSV scheme with damping, providing stability, optimal convergence, and superconvergence for linear scalar equations, and demonstrates its effectiveness through analysis and experiments.

Findings

The OFSV scheme is stable and converges optimally.

Damping controls oscillations without reducing accuracy.

Numerical results confirm robustness and effectiveness.

Abstract

In this paper, an oscillation-free spectral volume (OFSV) method is proposed and studied for the hyperbolic conservation laws. The numerical scheme is designed by introducing a damping term in the standard spectral volume method for the purpose of controlling spurious oscillations near discontinuities. Based on the construction of control volumes (CVs), two classes of OFSV schemes are presented. A mathematical proof is provided to show that the proposed OFSV is stable and has optimal convergence rate and some desired superconvergence properties when applied to the linear scalar equations. Both analysis and numerical experiments indicate that the damping term would not destroy the order of accuracy of the original SV scheme and can control the oscillations discontinuities effectively. Numerical experiments are presented to demonstrate the accuracy and robustness of our scheme.