Error analysis of the Crank–Nicolson SAV method for the Allen–Cahn equation on variable grids

Abstract

The variable time-stepping technique is powerful in capturing the multi-scale behaviors (e.g., the solution changes rapidly in certain regions of time) for the Allen–Cahn equation. Based on the scalar auxiliary variable (SAV) approach and the Crank–Nicolson schemes, we establish the unconditional energy stability and error estimates rigorously for the Allen–Cahn equation on variable grids. A numerical experiment is performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the convergence analysis for the SAV schemes on variable grids.