New Multi-implicit Space–Time Spectral Element Methods for Advection–Diffusion–Reaction Problems

Abstract

Novel multi-implicit space–time spectral element methods are described for approximating solutions to advection–diffusion–reaction problems characterized by multiple time scales. The new methods are spectrally accurate in space and time and they are designed to be easy to implement and robust. In other words, given an existing stable low order operator split method for approximating solutions to PDEs exhibiting multiple scales, the algorithms described in this article enable one to easily extend a low order method to be a robust space–time spectrally accurate method. In space, two spectrally accurate advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. In time, for the hyperbolic term(s), a low-order explicit I-stable building block time integration scheme is introduced in order to obtain a stable and efficient building block for the spectrally accurate space–time scheme. In this article, multiple spectrally accurate space discretization strategies, and multiple spectrally accurate time discretization strategies are compared to one another. It is found that all methods described are spectrally accurate with each method having distinguishing properties.