Energy dissipation laws of time filtered BDF methods up to fourth-order for the molecular beam epitaxial equation

Abstract

This report presents the time filtered BDF-k (FiBDF-k) methods up to fourth-order time accuracy for the molecular beam epitaxial equation with no-slope selection. The new (k+1)-order methods are developed by introducing an inexpensive post-filtering step to the variable-step BDF-k(k=1,2,3) methods. We show that the FiBDF-k methods are uniquely solvable and volume conservative. Some novel discrete gradient structures of the FiBDF-k formulas are derived such that we can build up the discrete energy dissipation laws for the associated time-steppings. Numerical examples are included to show the mesh-robustness and effectiveness of the variable-step methods especially in the long-time simulations.