High order accurate in time, fourth order finite difference schemes for the harmonic mapping flow

Abstract

In this paper, a fully discrete numerical scheme is proposed and analyzed for the harmonic mapping flow, with the fourth order spatial accuracy and higher than third order temporal accuracy. The fourth order spatial accuracy is realized via a long stencil finite difference, and the boundary extrapolation is implemented by making use of higher order Taylor expansion. Meanwhile, the high order (third or fourth order) temporal accuracy is based on a semi-implicit algorithm, which uses a combination of explicit Adams–Bashforth extrapolation for the nonlinear terms and implicit Adams–Moulton interpolation for the viscous diffusion term, with the corresponding integration formula coefficients. Both the consistency, linearized stability analysis and optimal rate convergence estimate (in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm) are provided. A few numerical examples are also presented in this article.