Dynamically adaptive simulations with minimal memory requirement-solving the shallow water equations using sierpins ki curves

We present an approach to the numerical simulation of dynamically adaptive problems on recursively structured adaptive triangular grids. The intended application is the simulation of oceanic wave propagation (e.g., tsunami simulation) based on the shallow water equations. For the required 2D dynamically adaptive discretization, we adopt a grid generation process based on recursive bisection of triangles along marked edges. The recursi ve grid generation may be described via a respective refinement tree, which is sequentialized according to a Sierpinski space-filling curve. This allows a storage scheme for the adaptive grid that requires only a minimal amount of memory. Moreover, the sequentialization and, hence, the locality properties induced by the space-filling curve are retained throughout adaptive refinement and coarsening of the grid. Conforming adaptive refinement and coarsening, as well as time-stepping techniques for time-dependent systems of partial differential equations, are implemented using an inherently cache-efficient processing scheme, which is based on the use of stacks and stream-like data structures and a traversal of the adaptively refined grid along the Sierpinski curve. We demonstrate the computational efficiency of the approach on the solution of a simplified version of the shallow water equations, for which we use a discontinuous Galerkin discretization. Special attention is paid to the memory efficiency of the implementation.