Exploring space-time adaptivity and parallel-in-time convergence for hyperbolic PDEs

Hans Johansen (Lawrence Berkeley National Laboratory)

We present an analysis of an adaptive space-time algorithm for hyperbolic partial differential equations. The spatial discretization we use is either explicit or based on implicitly-defined compact stencils, which require solving a linear system but can also have improved spectral properties. The space-time refinement uses nested regions with finer grid spacing/time step, that can be used to improve the error of solutions near steep gradients or material features. We explore several time integration options, including explicit, ADI, spectral deferred corrections, extrapolation, and exponential methods, and analyze the results in terms of spectral accuracy. Finally, we demonstrate a combined parallel-refinement-in-time algorithm and demonstrate why the spectral properties of operators in space and time must be considered together for optimal efficiency and convergence.