Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models

Abstract

We propose a family of novel, high-order numerical schemes for gradient flow models based on the scalar auxiliary variable (SAV) approach and name them the high-order scalar auxiliary variable (HSAV) methods. The proposed schemes are shown to reach arbitrarily high order in time while preserving the energy dissipation rate and thereby being unconditionally energy stable. When the HSAV strategy is applied to thermodynamically consistent gradient flow models, we arrive at semi-discrete high-order, unconditionally energy-stable schemes. We then employ the Fourier pseudospectral method in space to arrive at fully discrete unconditionally energy stable schemes. A few selected HSAV schemes are tested against three benchmark problems to demonstrate the accuracy, efficiency and unconditional energy stability of the schemes. The numerical results confirm the expected order of accuracy and robustness in much larger time steps than the low order SAV schemes.