Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods

TL;DR

This paper extends upwind summation-by-parts operators to nodal discontinuous Galerkin methods, demonstrating their construction on various grid points and analyzing their impact on robustness for compressible Euler equations.

Contribution

It introduces a broad class of USBP operators for nodal DG methods, including on Legendre-Gauss-Lobatto points, and evaluates their effectiveness in improving robustness without excessive dissipation.

Findings

USBP operators can be constructed on arbitrary grid points.

Combining flux vector splitting with DG-USBP enhances robustness.

Higher-order USBP operators offer less robustness improvement than lower-order ones.

Abstract

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously -- sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin--Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes...