Explicit and implicit error inhibiting schemes with post-processing

Abstract

High order implicit-explicit (IMEX) methods are often desired when evolving the solution of an ordinary differential equation that has a stiff part that is linear and a non-stiff part that is nonlinear. This situation often arises in semi-discretization of partial differential equations and many such IMEX schemes have been considered in the literature. The methods considered usually have a a global error that is of the same order as the local truncation error. More recently, methods with global errors that are one order higher than predicted by the local truncation error have been devised [18,28,5]. In prior work [6,7] we investigated the interplay between the local truncation error and the global error to construct explicit and implicit error inhibiting schemes that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error, and which can be post-processed to obtain a solution which is two orders higher than expected. In this work we extend our error inhibiting with post-processing framework introduced in [6] and further in [7] to a class of additive general linear methods with multiple steps and stages. We provide sufficient conditions under which these methods with local truncation error of order p will produce solutions of order p + 1, which can be post-processed to order p + 2, and describe the construction of one such post-processor. We apply this approach to obtain implicit-explicit (IMEX) methods with multiple steps and stages. We present some of our new IMEX methods and show their linear stability properties, and investigate how these methods perform in practice on some numerical test cases.

A Holistic Algorithmic Approach to Improving Accuracy, Robustness, and Computational Efficiency for Atmospheric Dynamics

“…Also, the speed-ups obtained tend to be low compared to the additional amount of resources required. Recently, a new and simple parallel-in-time technique has emerged that allows the multiple stages of computation to be computed concurrently [6]. One could think of these schemes as block-stepped multistage integrators that incorporate multistep information as well.…”

A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD

“…An A-stable third order method that is based on the implicit Euler method can be obtained by using the error inhibiting approach presented in [EIS20]. In this formulation, we retain previous stages as well as previous steps, to create a method of the form…”

“…Example 3: Looking at the pre-and post-filtering process as a general linear method (GLM) also opens the door to creating methods that have the error inhibiting properties described in [EIS20]. An example of this is the EIS method IE-EIS-3 which we will describe in 4.1.4:…”

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