A dual-pairing summation-by-parts finite difference framework for nonlinear conservation laws

TL;DR

This paper introduces a dual-pairing SBP finite difference and discontinuous Galerkin framework that enhances the robustness and accuracy of high-order numerical methods for nonlinear conservation laws, ensuring entropy consistency and effective shock capturing.

Contribution

It presents a novel dual-pairing SBP framework combining FD and DG methods with upwind properties, improving stability and accuracy for nonlinear hyperbolic PDEs.

Findings

Framework achieves entropy consistency.

Operators preserve SBP property and are upwind.

Enhanced robustness in capturing discontinuities.

Abstract

Robust and convergent high-order numerical methods for solving partial differential equations are highly attractive due to their efficiency on modern and next-generation hardware architectures. However, designing such methods for nonlinear hyperbolic conservation laws remains a significant challenge. In this work, we introduce a framework based on dual-pairing (DP) and upwind summation-by-parts (SBP) finite difference (FD) and discontinuous Galerkin (DG) finite element methods, aimed at achieving accurate and robust numerical approximations of nonlinear conservation laws. The framework ensures entropy consistency and features an intrinsic high-order accurate filter designed to detect and resolve regions where the solution is poorly captured or discontinuities are present. The DP SBP FD/DG operators form a dual pair of discrete derivative operators that collectively preserve the SBP property. Furthermore, these operators are constructed to be upwind, allowing them to incorporate dissipation within the elements themselves.This contrasts with traditional SBP and collocated DG spectral element methods, which typically induce dissipation solely through numerical fluxes at element interfaces. Our framework facilitates the systematic combination of DP SBP FD/DG operators with skew-symmetric and upwind flux splitting techniques. This integration enables the development of robust, high-order accurate schemes for nonlinear hyperbolic conservation laws.