An explicitness-preserving IMEX-split multiderivative method

Abstract

In the last decade, multi-derivative schemes for ordinary differential equations (ODE), i.e., schemes not only using the ODE's flux, but also derivatives thereof, have seen renewed interest in various applications. In [Schütz and Seal, Applied Numerical Mathematics, 160, 2021], the authors have introduced a two-derivative method for applications where a clear distinction can be made between stiff (to be treated implicitly) and non-stiff parts (to be treated explicitly); they hence obtained a two-derivative IMEX (im-plicit/explicit) scheme. In many applications, it is important that the non-stiff part is only treated explicitly. Think, e.g., of applications where the non-stiff part contains all the nonlinearities and should for efficiency reasons hence only be treated explicitly. However, this non-stiff part showed up in the second derivative of the stiff term and was hence treated implicitly as well. In this work, we further develop the algorithm from Schütz and Seal in such a way that the explicit part will be treated explicitly only. We define the algorithm, show that it preserves the asymptotic behavior of certain stiff equations, and investigate it numerically. We found that the modification we make has no negative effect on the convergence behavior of the scheme.