A Time Two-grid Algorithm Based On Finite Difference Method For The Two-dimensional Nonlinear Time-fractional Mobile/Immobile Transport Model

In this work,we consider a two-dimensional nonlinear time-fractional mobile/immobile transport model,which is solved via proposing a time two-grid algorithm based on the finite differ-ence(FD)method.We establish the problem as a nonlinear fully-discrete FD system,where the time derivative is discretized via the second-order backward differentiation formula(BDF),the Caputo fractional derivative is processed by the L1 discretization formula,and the spatial derivative is approximated via the central difference formula.In order to solve the nonlinear FD system more effectively,a time two-grid algorithm is proposed.The algorithm includes two steps:first,the nonlinear FD system is solved on the coarse grid by nonlinear iterations;second,use interpolation for the solutions of the coarse grid,which provide the initial value for the fine grid,thus we can solve the linearized FD system on a fine grid.For the time two-grid FD scheme,we obtain the stability and conver-gence under the L~2norm.In addition,the numerical results are consistent with the theoretical analysis.Meanwhile,numerical ex-periments illustrate that the two-grid FD method can reduce more computing costs than the general FD method for solving nonlinear FD systems.The structure of this paper is organized as follows:In Chapter1,we introduce the research background of the two-dimensional nonlinear time-fractional mobile/immobile transport model,the previous researches on the two-grid algorithm and the advantages of this method,and the main content of the research.In Chapter2,we introduce some notations,important definitions and aux-iliary lemmas.In Chapter 3,for the two-dimensional nonlinear time-fractional mobile/immobile transport model,a time two-grid FD scheme is constructed.In Chapter 4,the stability and conver-gence analysis of the time two-grid FD scheme are carried out.In Chapter 5,some numerical experiments are provided,whose results are compared with those of the general nonlinear implicit difference scheme.In Chapter 6,we give summary and outlook of this article.