An adaptive step size controller for iterative implicit methods

TL;DR

This paper introduces an adaptive step size controller for iterative implicit methods that reduces computational cost per step, improving efficiency across various differential equations by allowing rapid step size adjustments.

Contribution

It proposes a novel step size selection strategy that adaptively minimizes computational effort per unit time step, especially when iterative solvers are used.

Findings

Significant performance improvements demonstrated on multiple differential equations.

The approach enables rapid and effective step size adjustments.

Enhanced efficiency over traditional smooth step size sequences.

Abstract

The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used. In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion-advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen--Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.