An asymptotic preserving semi-implicit multiderivative solver

Abstract

In this work we construct a multiderivative implicit-explicit (IMEX) scheme for a class of stiff ordinary differential equations (ODEs). Our solver is high-order accurate and has an asymptotic preserving (AP) property for a large class of singularly perturbed ODEs. In this context, the AP property means that the singular limit is discretely preserved when a stiff parameter ε goes to zero. The proposed method is based upon a two-derivative backward Taylor series base solver, which we show has an AP property. Higher order accuracies are found by iterating the result over a high-order multiderivative interpolant of the right hand side function, which we again prove has an AP property. Theoretical results showcasing the asymptotic consistency as well as the high-order accuracy of the solver are presented. In addition, an extension of the solver to an arbitrarily split right hand side function is also offered. Numerical results for a collection of standard test cases from the literature are presented that support the theoretical findings of the paper.