Error Estimation for Multi-Stage Runge-Kutta IMEX Schemes

TL;DR

This paper develops a posteriori error estimates for multi-stage IMEX schemes using a nodally equivalent finite element approach and adjoint-based methods, enabling detailed error decomposition for quantities of interest.

Contribution

It introduces a novel error estimation framework for IMEX schemes by linking them to finite element methods and applying adjoint techniques for detailed error analysis.

Findings

Effective error decomposition into multiple components

Enhanced accuracy in error estimation for IMEX schemes

Framework applicable to a wide range of problems

Abstract

Implicit-Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity of interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. The use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity of interest.