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

Zhuoyun Li, Kailiang Wu

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.

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

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.
Paper Structure (35 sections, 27 theorems, 116 equations, 8 figures, 4 tables)

This paper contains 35 sections, 27 theorems, 116 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. SV and OESV methods
  3. Semi-discrete SV formulation
  4. Runge--Kutta SV method
  5. OESV method
  6. Understanding SV from a DG perspective
  7. Preliminary
  8. DG representation of semi-discrete SV method
  9. DG representations of fully-discrete RKSV and OESV methods
  10. Stability and optimal error estimates
  11. Main results
  12. Proof of \ref{['thm:stability-RKSV']}
  13. Temporal difference operator
  14. Energy equality and inequality
  15. Estimates for source terms
  16. ...and 20 more sections

Key Result

Lemma 3.2

\newlabellemma:Mstar-difference0 For any $v \in L^2(\Omega)$, if there exists $V \in L^2(\Omega)$ such that $V_x = v$, then

Figures (8)

  • Figure 1: OESV solutions (top) at $t=1.8$ and their cut along $y=-0.25$ (bottom) for \ref{['ex:2Dadvec2']}. From left to right: $\mathbb Q^1$, $\mathbb Q^2$, and $\mathbb Q^3$ approximations.
  • Figure 1: Shock reflection problem simulated by $\mathbb{Q}^2$-based OESV scheme.
  • Figure 2: Densities of two Riemann problems at $t=1.3$ computed by OESV scheme.
  • Figure 2: Supersonic flow problem simulated by $\mathbb{Q}^2$-based OESV scheme.
  • Figure 3: Densities of \ref{['ex:1Deuler3']} computed by the OESV scheme.
  • ...and 3 more figures

Theorems & Definitions (64)

  • Definition 3.1: Cao2021AnalysisOS
  • Lemma 3.2: Cao2021AnalysisOS
  • Proposition 3.3
  • Theorem 3.5
  • Proof 1
  • Remark 3.6
  • Remark 3.7
  • Proposition 3.8
  • Definition 3.9
  • Proposition 3.10
  • ...and 54 more