A parallel-in-time, implicit/explicit multiderivative solver

Jochen Schuetz (Hasselt University)
David Seal (US Naval Academy)

In this talk, we present a novel scheme for a class of extremely stiff ordinary differential equations.

Frequently, singularly perturbed differential equations can be split into stiff and non-stiff parts. If so, a flux splitting can typically be constructed, and stiff parts are treated implicitly-in-time for stability, while the other parts are treated explicitly for efficiency. This treatment has been termed IMEX. It is, amongst others, very successful for the computation of solutions to relaxation problems and low-Mach fluid flow equations.

To the best of our knowledge, the algorithm that we present in this talk is the first attempt ever to combine IMEX methods with so-called multiderivative methods. For multiderivative methods and an ODE of form y’(t) = f(y(t)), say, the algorithm does not only take y’(t) = f(y) into account, but also y’’(t) = f’(y) * f(y); this typically results in high-order methods with very few storage requirements. The resulting method that we present here is provably stable for prototypical equations and can be extended to partial differential equations. The scheme is of predictor-corrector type which makes it easily amenable to temporal parallelisation.

In this talk, we will present both analytical and numerical results for the use of the method with ordinary differential equations, including its use parallel-in-time. Subsequently, we will show how to extend the method to the low-Mach Euler equations.

[1] D. Seal and J. Schütz. An asymptotic preserving semi-implicit multiderivative solver. CMAT Preprint UP-19-09, http://www.uhasselt.be/Documents/CMAT/Preprints/2019/UP1909.pdf , 2019.