Nonuniform Time-stepping Approaches And High-order Local Absorbing Boundarys Condition On Unbounded Strips For Subdiffusion Problems

The study of this paper is two-fold.Firstly,we reviewed the recent development and gave the outlook about nonuniform time-stepping methods in subdiffusion equation.Second-ly,we consider the two-dimension subdiffusion problems on unbounded strips,and design a class of high-order artificial boundary conditions(ABCs).Nonuniform time-stepping methods are promising for Caputo reaction-subdiffusion problems because they would be simple and effectiveness in resolving the initial singularity and other nonlinear behaviors occurred away from the initial time.Compared with traditional local methods for the first-order derivative,the numerical analysis for nonlocal time-stepping schemes on nonuniform time meshes are chal-lenging due to the convolution integral(nonlocal)form of fractional derivative.We develop a general framework for the stability and convergence analysis with three tools:a family of com-plementary discrete convolution kernels,a discrete fractional Gronwall inequality(DFGI)and a global(convolutional)consistency analysis,which is not limited to a specific time mesh by building a convolution structure of local truncation error.It seems that the present techniques are extendable to the variable-order,distributed-order diffusion equations and other nonlocal-in-time diffusion problems.We reduce the two-dimension subdiffusion problems on unbounded strips to initial boundary value problems by deriving high-order approximate artificial boundary conditions.After that,the IB VPs with our high-order ABCs are proved to be stable in the L2-norm.And unconditionally stable schemes are constructed to numerically solve the IB VPs by using L1 approximation to discretize the temporal derivative and using finite difference methods to discretize the spatial derivative.Finally,numerical examples are provided to demonstrate the effectiveness of the proposed schemes in this paper.