A Posteriori Error Analysis of Imex Multi-step Time Integration Methods for Advection-diffusion-reaction Equations

Implicit Explicit (IMEX) schemes are an important and widely used class of time integration methods for both parabolic and hyperbolic partial differential equations. We develop accurate a posteriori error estimates for a user-defined quantity of interest for two classes of multistep IMEX schemes for advection-diffusion-reaction problems. The analysis proceeds by recasting the IMEX schemes into a variational form suitable for a posteriori error analysis employing adjoint problems and computable residuals. The a posteriori estimates quantify distinct contributions from various aspects of the spatial and temporal discretizations, and can be used to evaluate discretization choices. Numerical results are presented that demonstrate the accuracy of the estimates for a representative set of problems.