Scalar hyperbolic PDE simulations and coupling strategies

We report on grid-adaptive, multi-dimensional simulations for hyperbolic PDEs, with a deliberate focus on the analytically tractable scalar case. Motivated by recent efforts towards multi-physics simulation strategies, we investigate a variety of coupling strategies for numerically solving hyperbolic partial differential equations (PDEs). We use adaptive mesh refinement in combination with shock-capturing spatio-temporal discretizations, and first present accuracy and validation tests on both smooth and shock-dominated evolutions. To investigate the feasibility of coupling means for multi-physics simulations, we then introduce new reference tests where spatially different flux prescriptions require coupling strategies across the domains where local advection, Burgers or nonconvex behavior is imposed. For these nonlinear single scalar equations, we can illustrate and analytically explain the evolutions obtained when handling cases where fluxes differ in spatially non-overlapping regions. We discuss both conservative and non-conservative ways of coupling across the region boundaries. When coupling scalar conservation laws where the characteristic speeds change discontinuously across the regional boundaries, these two strategies yield differing, but mathematically sound, solution behaviors. Their relevance for multi-physics simulations where one couples systems of (hyperbolic) PDEs is discussed.