Unconditionally energy stable and second-order accurate one-parameter ESAV schemes with non-uniform time stepsizes for the functionalized Cahn-Hilliard equation

Abstract

This paper studies linear and unconditionally energy stable schemes for the functionalized Cahn-Hilliard (FCH) equation. Such schemes are built on the exponential scalar auxiliary variable (ESAV) approach and the one-parameter time discretizations as well as the extrapolation for the nonlinear term, and can arrive at second-order accuracy in time. It is shown that the derived schemes are uniquely solvable and unconditionally energy stable by using an algebraic identity derived by the method of undetermined coefficients. Importantly, such one-parameter ESAV schemes are extended to those with non-uniform time stepsizes, which are also shown to be unconditionally energy stable by an analogous algebraic identity. The energy stability results can be easily extended to the fully discrete schemes, where the Fourier pseudo-spectral method is employed in space. Moreover, based on the derived schemes with non-uniform time stepsizes, an adaptive time-stepping strategy is introduced to improve the computational efficiency for the long time simulations of the FCH equation. Several numerical examples are conducted to validate the computational accuracy and energy stability of our schemes as well as the effectiveness and computational efficiency of the derived adaptive time-stepping algorithm.