Graph-based methods for hyperbolic systems of conservation laws using discontinuous space discretizations, Part I: building blocks

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.