Efficient High-Order Backward Difference Formulae for Cahn-Hilliard Equation with the Gradient Flow in $H^{−α}$

Abstract

High-order schemes enable simulating multi-time-scale problems with relatively large time step sizes for fairly accurate solutions. An unified high-order implicit BDF-$q$ $(q = 3,4,5)$ scheme is developed for the Cahn-Hilliard equation with the gradient flow in $H^{−α},α∈(0,1],$ including the classical case and its fractional variants. Introducing the discrete gradient structure, the resulting implicit BDF-$q$ scheme is presented to satisfy the discrete energy dissipation law, which is actually compatible with the one for the classical case as the order of the fractional Laplacian $α$ approaches 1. The $L^2$ norm error estimate for the BDF-$q$ scheme is rigorously proved by developing a discrete Young-type convolution inequality to deal with the nonlinear term along with the fractional Laplacian. Further, the high-order BDF-$q$ scheme is shown to be less time-consuming compared to the variable-step BDF-2 scheme, while the BDF-5 scheme reduces the CPU time in long-time simulation of coarsening dynamics by almost 80%. Numerical examples also demonstrate that high-order schemes are deemed appealing for long-time slow evolution, while variable-step scheme exhibits more flexibility during phase separation at initial state. In light of this, the variable-step BDF-$q$ scheme utilizing the adaptive time-stepping strategy is implemented to capture both the rapid and slow evolutions of the solutions efficiently and accurately even in high dimensions.