Highly accurate, linear, and unconditionally energy stable large time-stepping schemes for the Functionalized Cahn–Hilliard gradient flow equation

Abstract

This paper aims to develop a class of large time-stepping schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation. By using the Invariant Energy Quadratization (IEQ) approach, we successfully construct several highly efficient linear schemes that allow large time step sizes to be used in computations. All the developed schemes with a fixed time step size are proved to be unconditionally uniquely solvable and unconditionally energy stable in theory. Moreover, we show that the obtained semi-discrete systems can be solved efficiently by using the preconditioned Krylov subspace methods, since all the linear operators of these systems are proved to be symmetric and positive definite. Finally, several numerical experiments are carried out to verify the accuracy, efficiency and energy stability of the developed IEQ schemes. We observe that our IEQ schemes are more accurate than the Scalar Auxiliary Variable (SAV) schemes. This can be attributed to the fact that the IEQ method can preserve the operator structures of the original equation as much as possible.