Stable Spectral-Volume Methods

TL;DR

This work addresses instability in high-order Spectral-Volume methods for hyperbolic conservation laws by adapting Dafermos' entropy rate criterion from the DG context. It develops a discrete, conservative, positive filter, grounded in a heat-equation generator, to enforce entropy dissipation at SV boundaries and guarantee the entropy inequality. The stabilized SV scheme demonstrates reduced spurious oscillations and preserved sharp discontinuities across linear advection, Burgers’ equation, and Euler shock-tube problems, with convergence analysis revealing fourth- to fifth-order behavior despite using RK3 in time. The approach offers a robust, high-order, entropy-stable SV framework with practical implications for simulations of compressible flows and other hyperbolic systems.

Abstract

A novel approach for the stabilization of the Spectral-Volume (SV) method based on Dafermos' entropy rate criterion is presented. The method is an adaption of an already existing approach for the stabilization of the Discontinuous-Galerkin (DG) method. It employs the same estimates for the maximal possible entropy dissipation rate as the DG version. However, a new way to compute the discrete conservative filter had to be derived due to the differences of the underlying schemes. The resulting modified SV scheme even satisfies the entropy inequality. Tests are carried out for Burgers' equation and for the Euler equations of gas dynamics.

Figures (16)

• Figure 1: Partition into spectral volumes and further subdivision into control volumes.
• Figure 2: Polynomial reconstruction: (a) Cell averages, (b) Reconstruction polynomial, (c) Reconstructed boundary values. One can see the reconstruction within one spectral volume that is divided into four CVs. Initially cell averages are given for all four control volumes. In a first step the coefficients of the reconstruction polynomial are calculated via a multiplication of these cell averages by $\mathbf{M^{-1}}$. Subsequently, the reconstruction polynomial is evaluated at all CV boundaries via a multiplication of the vector of coefficients by the matrix $\mathbf{A}$. In practice, the two steps are of course combined in one multiplication of the given cell averages by the reconstruction matrix $\mathbf{C}$.
• Figure 3: Numerical flux: (a) Reconstructed boudary values, (b) Numerical flux The computation of the numerical flux is depicted for one spectral volume that is divided into four CVs, with half of the adjacent SVs also visible. Initially, reconstructed values of $u$ at the CV boundaries are provided, calculated from the cell averages $\bar{u}_{i,j}$ as explained in the preceding chapter. At SV boundaries, two distinct values of $u$ are obtained-one resulting from the reconstruction in the adjacent SV on the left and another from the reconstruction in the adjacent SV on the right. The ensuing Riemann problem is solved using an approximate Riemann solver. Consequently, at SV boundaries, the numerical flux is computed using the local Lax-Friedrichs flux $f^{LLF}$ indicated by red squares. At inner CV boundaries, only one reconstructed value of $u$ is obtained, as the reconstruction polynomial remains consistent throughout the entire spectral volume. Thus, at inner CV boundaries, the analytical flux $f$ can be directly applied indicated by blue circles. In summary, this case-based definition distinctly determines the numerical flux $f^{*}$ at every CV boundary.
• Figure 4: Solution at $t = 1.0$
• Figure 5: $L^{2}$-norm of the numerical solution over time.

Theorems & Definitions (11)

• Theorem 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 2.10