Graph-based methods for hyperbolic systems of conservation laws using discontinuous space discretizations

TL;DR

This work develops a graph-based discontinuous Galerkin framework for hyperbolic systems of conservation laws, integrating boundary conditions directly into the theory and implementation. It combines a robust first-order invariant-set preserving scheme with a high-order entropy-viscosity approach and a convex limiting strategy that uses local bounds, all formulated on a single stencil. The resulting method delivers provable discrete entropy inequalities and invariant-set preservation, while achieving high-order accuracy for smooth problems and reliable shock-capturing for discontinuities; boundary conditions are handled in a physically consistent, component-wise manner, including sub-cell limiting. Numerical tests spanning smooth vortices, rarefaction waves, and strong shocks demonstrate proper convergence rates and robustness, with a high-fidelity Mach 3 cylinder flow illustrating practical applicability on complex geometries.

Abstract

We present a graph-based numerical method for solving hyperbolic systems of conservation laws using discontinuous finite elements. This work fills important gaps in the theory as well as practice of graph-based schemes. In particular, four building blocks required for the implementation of flux-limited graph-based methods are developed and tested: a first-order method with mathematical guarantees of robustness; a high-order method based on the entropy viscosity technique; a procedure to compute local bounds; and a convex limiting scheme. Two important features of the current work are the fact that (i) boundary conditions are incorporated into the mathematical theory as well as the implementation of the scheme. For instance, the first-order version of the scheme satisfies pointwise entropy inequalities including boundary effects for any boundary data that is admissible; (ii) sub-cell limiting is built into the convex limiting framework. This is in contrast to the majority of the existing methodologies that consider a single limiter per cell providing no sub-cell limiting capabilities. From a practical point of view, the implementation of graph-based methods is algebraic, meaning that they operate directly on the stencil of the spatial discretization. In principle, these methods do not need to use or invoke loops on cells or faces of the mesh. Finally, we verify convergence rates on various well-known test problems with differing regularity. We propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

Figures (1)

• Figure 1: Temporal snapshot at time $t=4.0$ of the density profiles of a supersonic Mach 3 flow past a cylinder. Computed for increasing polynomial degree: (a) $\mathbb{Q}^1$, (b) $\mathbb{Q}^2$, (c) $\mathbb{Q}^3$. The density is visualized on a rainbow colormap to highlight discontinuities.

Theorems & Definitions (25)

• Remark 2.5: Choice of basis functions
• Lemma 3.1: Partition of unity property
• Proof 1
• Lemma 3.2
• Proof 2
• Lemma 3.3: Skew symmetric form
• Corollary 3.4: Total balance
• Remark 3.5
• Lemma 3.6: Convex reformulation
• Proof 3