On the SAV-DG method for a class of fourth order gradient flows

TL;DR

This paper introduces an efficient hybrid SAV-DG method for fourth order gradient flows that reduces computational complexity from quadratic to linear, maintaining high accuracy and energy stability.

Contribution

The paper develops a pre-evaluation procedure for the auxiliary variable in SAV-DG schemes, significantly improving computational efficiency for solving fourth order gradient flows.

Findings

Reduced computational complexity from O(N^2) to O(N) using conjugate gradient solver.

The hybrid SAV-DG method achieves high accuracy and energy stability.

Compared favorably with solving the full augmented system.

Abstract

For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty-free discontinuous Galerkin (DG) spatial discretization leads to SAV-DG schemes. These schemes are linear and shown unconditionally energy stable. But the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre-evaluate the auxiliary variable in the piecewise polynomial space. As a result, the computational complexity of $O(\mathcal{N}^2)$ reduces to $O(\mathcal{N})$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV-DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV-DG schemes.