Graph-Based Methods for Hyperbolic Systems of Conservation Laws Using Discontinuous Space Discretizations (Communications in Computational Physics, 2025)
We present a graph-based numerical method for solving hyperbolic systemsof conservation laws using discontinuous finite elements. This work fills importantgaps in the theory as well as practice of graph-based schemes. In particular, fourbuilding blocks required for the implementation of flux-limited graph-based methods are developed and tested: a first-order method with mathematical guarantees ofrobustness; a high-order method based on the entropy viscosity technique; a procedure to compute local bounds; and a convex limiting scheme. Two important featuresof the current work are the fact that (i) boundary conditions are incorporated intothe mathematical theory as well as the implementation of the scheme. For instance,the first-order version of the scheme satisfies pointwise entropy inequalities includingboundary effects for any boundary data that is admissible; (ii) sub-cell limiting is builtinto the convex limiting framework. This is in contrast to the majority of the existingmethodologies that consider a single limiter per cell providing no sub-cell limiting capabilities. From a practical point of view, the implementation of graph-based methodsis 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 ofthe mesh. Finally, we verify convergence rates on various well-known test problemswith differing regularity. We propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.
Keywords: Discontinuous finite elements; graph-based formulation; hyperbolic systems; invariant sets; convex limiting; boundary conditions.