Splitting Schemes for ODEs with Goal-Oriented Error Estimation

TL;DR

This paper introduces a hybrid goal-oriented error estimator for ODEs that combines dynamic iteration and dual weighted residual methods, enabling adaptive discretization and efficient solution refinement.

Contribution

It presents a novel combined error estimator for ODEs that balances dynamic iteration and discretization errors, facilitating adaptive mesh refinement and flexible time discretization.

Findings

Adaptive approach outperforms uniform refinement in numerical experiments.

The estimator effectively balances iteration and discretization errors.

Flexible discretization matches multiple time scale requirements.

Abstract

We present a hybrid a-priori/a-posteriori goal oriented error estimator for a combination of dynamic iteration-based solution of ordinary differential equations discretized by finite elements. Our novel error estimator combines estimates from classical dynamic iteration methods, usually used to enable splitting-based distributed simulation, and from the dual weighted residual method to be able to evaluate and balance both, the dynamic iteration error and the discretization error in desired quantities of interest. The obtained error estimators are used to conduct refinements of the computational mesh and as a stopping criterion for the dynamic iteration. In particular, we allow for an adaptive and flexible discretization of the time domain, where variables can be discretized differently to match both goal and solution requirements, e.g. in view of multiple time scales. We endow the scheme with efficient solvers from numerical linear algebra to ensure its applicability to complex problems. Numerical experiments compare the adaptive approach to a uniform refinement.