Explicit adaptive time stepping for the Cahn-Hilliard equation by exponential Krylov subspace and Chebyshev polynomial methods

TL;DR

This paper introduces an explicit exponential time integration method for the Cahn-Hilliard equation, utilizing Krylov subspace and Chebyshev polynomial techniques, optimized for parallel computing environments.

Contribution

It presents a novel exponential Krylov subspace method and compares it with a Chebyshev polynomial scheme for efficient explicit time stepping.

Findings

The exponential Krylov method performs well on parallel architectures.

Both methods are effective with constant and adaptive time steps.

The proposed methods avoid solving linear systems, enhancing efficiency.

Abstract

The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation, especially on parallel and hybrid supercomputers. In this paper, we propose an exponential time integration method for the Cahn-Hilliard equation and describe its efficient Krylov subspace based implementation. We compare the method to a Chebyshev polynomial local iteration modified (LIM) time stepping scheme. Both methods are explicit (i.e., do not involve linear system solution) and tested with both constant and adaptively chosen time steps.