Developing a large time step, robust, and low communication multi-moment PDE integration scheme for exascale applications

The Boundary Averaged Multi-moment Constrained finite-Volume (BA-MCV) method is derived, explained, and evaluated for 1-D transport to assess accuracy, maximum stable time step (MSTS), oscillations for discontinuous data, and parallel communication burden. The BA-MCV scheme is altered from the original MCV scheme to compute the updates of point wise cell boundary derivatives entirely locally. Then it is altered such that boundary moments are replaced with the interface upwind value. The scheme is stable at a maximum stable CFL (MSCFL) value of one no matter how high-order the scheme is, giving significantly larger time steps than Galerkin methods, for which the MSCFL decreases nearly quadratically with increasing order. The BA-MCV method is compared against a SE method at varying order, both using the ADER-DT time discretization. BA-MCV error for a sine wave was comparable to the same order of accuracy for a SE method. The resulting large time step, multi-moment, low communication scheme is of great interest for exascale architectures.