Spectral Volume from a DG perspective: Oscillation Elimination, Stability, and Optimal Error Estimates

TL;DR

This paper presents a spectral volume method reformulated from a DG perspective, introduces an oscillation-eliminating technique, and provides theoretical stability and error estimates, validated by numerical experiments for hyperbolic conservation laws.

Contribution

It extends the spectral volume method within a DG framework, develops a new oscillation-eliminating technique, and establishes stability and optimal error estimates for high-resolution hyperbolic PDE simulations.

Findings

The OESV method effectively suppresses spurious oscillations.

Stability of RKSV schemes depends on an upwind condition.

Numerical results confirm theoretical accuracy and robustness.

Abstract

The discontinuous Galerkin (DG) method and the spectral volume (SV) method are two widely-used numerical methodologies for solving hyperbolic conservation laws. In this paper, we demonstrate that under specific subdivision assumptions, the SV method can be represented in a DG form with a different inner product. Building on this insight, we extend the oscillation-eliminating (OE) technique, recently proposed in [M. Peng, Z. Sun, and K. Wu, {\it Mathematics of Computation}, https://doi.org/10.1090/mcom/3998], to develop a new fully-discrete OESV method. The OE technique is non-intrusive, efficient, and straightforward to implement, acting as a simple post-processing filter to effectively suppress spurious oscillations. From a DG perspective, we present a comprehensive framework to theoretically analyze the stability and accuracy of both general Runge-Kutta SV (RKSV) schemes and the novel OESV method. For the linear advection equation, we conduct an energy analysis of the fully-discrete RKSV method, identifying an upwind condition crucial for stability. Furthermore, we establish optimal error estimates for the OESV schemes, overcoming nonlinear challenges through error decomposition and treating the OE procedure as additional source terms in the RKSV schemes. Extensive numerical experiments validate our theoretical findings and demonstrate the effectiveness and robustness of the proposed OESV method. This work enhances the theoretical understanding and practical application of SV schemes for hyperbolic conservation laws, making the OESV method a promising approach for high-resolution simulations.