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In this paper, we propose a novel family of high-order numerical schemes forthe gradient flow models based on the scalar auxiliary variable (SAV) approach,which is named the high-order scalar auxiliary variable (HSAV) method. Thenewly proposed schemes could be shown to reach arbitrarily high order in timewhile preserving the energy dissipation law without any restriction on the timestep size (i.e., unconditionally energy stable). The HSAV strategy is rathergeneral that it does not depend on the specific expression of the effectivefree energy, such that it applies to a class of thermodynamically consistentgradient flow models arriving at semi-discrete high-order energy-stableschemes. We then employ the Fourier pseudospectral method for spatialdiscretization. The fully discrete schemes are also shown to be unconditionallyenergy stable. Furthermore, we present several numerical experiments on severalwidely-used gradient flow models, to demonstrate the accuracy, efficiency andunconditionally energy stability of the HSAV schemes. The numerical resultsverify that the HSAV schemes can reach the expected order of accuracy, and itallows a much larger time step size to reach the same accuracy than thestandard SAV schemes.

Arbitrarily High-order Unconditionally Energy Stable Schemes for Gradient Flow Models Using the Scalar Auxiliary Variable Approach