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

Zhuoyun Li, Kailiang Wu
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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.
从 DG 角度看频谱量:振荡消除、稳定性和最佳误差估计
非连续伽勒金(DG)方法和谱体积(SV)方法是求解双曲守恒定律的两种广泛使用的数值方法。在本文中,我们证明了在特定的细分假设下,SV 方法可以通过不同的inner product 以 DG 形式表示。在此基础上,我们扩展了最近在[M. Peng, Z. Sun, and K. S. and M. P.Peng, Z. Sun, and K. Wu, {\itMathematics of Computation}, https://doi.org/10.1090/mcom/3998] 中提出的振荡消除(OE)技术,发展出一种新的全离散 OESV 方法。OE 技术非侵入式、高效且易于实现,可作为一种简单的后处理滤波器来有效抑制杂散振荡。从 DG 的角度,我们提出了一个综合框架,从理论上分析了一般 Runge-Kutta SV (RKSV) 方案和新型 OESV 方法的稳定性和准确性。对于线性平流方程,我们对完全离散的 RKSV 方法进行了能量分析,确定了对稳定性至关重要的上风条件。此外,我们还为 OESV 方案建立了最优误差估计,通过误差分解克服了非线性挑战,并将 OEprocedure 视为 RKSV 方案中的附加源项。广泛的数值实验验证了我们的理论发现,并证明了所提出的 OESV 方法的有效性和鲁棒性。这项工作增强了对双曲守恒定律 SV 方案的理论理解和实际应用,使 OESV 方法成为高分辨率模拟的一种有前途的方法。
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