Optimal adaptive implicit time stepping

TL;DR

This paper introduces a mathematically guaranteed optimal adaptive implicit time stepping algorithm that achieves the best possible convergence rate, advancing the theoretical understanding of adaptive time integration.

Contribution

It proposes a new adaptive time stepping method based on recent adaptive mesh refinement advances, with proven optimal convergence guarantees.

Findings

Algorithm achieves optimal error convergence rate.

Compatible with black-box time stepping schemes.

Provides a complete theoretical framework for adaptive time stepping.

Abstract

We revisit adaptive time stepping, one of the classical topics of numerical analysis and computational engineering. While widely used in application and subject of many theoretical works, a complete understanding is still missing. Apart from special cases, there does not exist a complete theory that shows how to choose the time steps such that convergence towards the exact solution is guaranteed with the optimal convergence rate. In this work, we use recent advances in adaptive mesh refinement to propose an adaptive time stepping algorithm that is mathematically guaranteed to be optimal in the sense that it achieves the best possible convergence of the error with respect to the number of time steps, and it can be implemented using a time stepping scheme as a black box.