Unconditionally energy stable DG schemes for the Swift-Hohenberg equation

TL;DR

This paper develops fully discrete discontinuous Galerkin schemes for the Swift-Hohenberg equation that guarantee unconditional energy stability, are efficient, and produce accurate pattern formation results.

Contribution

It introduces a novel IEQ-DG method for fourth order gradient flow problems, ensuring unconditional energy stability and efficiency without iterative solvers.

Findings

Schemes are proven to be unconditionally energy stable.

Numerical examples demonstrate high accuracy and efficiency.

Method successfully captures complex pattern formations.

Abstract

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.