Energy stable schemes for gradient flows based on novel auxiliary variable with energy bounded above

TL;DR

This paper introduces a novel auxiliary variable method for gradient flows that guarantees energy stability without bounded below restrictions, ensuring positivity of the functional and demonstrating improved accuracy and efficiency through numerical simulations.

Contribution

The proposed NAEV method removes the bounded below restriction present in previous approaches, guaranteeing positivity and unconditional energy stability for gradient flow schemes.

Findings

Unconditional energy stability proven for all semi-discrete schemes

NAEV guarantees positivity of the energy functional

Numerical simulations confirm stability and accuracy

Abstract

In this paper, we consider a novel auxiliary variable method to obtain energy stable schemes for gradient flows. The auxiliary variable based on energy bounded above does not limited to the hypothetical conditions adopted in previous approaches. We proved the unconditional energy stability for all the semi-discrete schemes carefully and rigorously. The novelty of the proposed schemes is that the computed values for the functional in square root are guaranteed to be positive. This method, termed novel auxiliary energy variable (NAEV) method does not consider any bounded below restrictions any longer. However, these restrictions are necessary in invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches which are very popular methods recently. This property of guaranteed positivity is not available in previous approaches. A comparative study of classical SAV and NAEV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.