Alternating Direction Implicit Finite Difference Method For Several High-Dimensional Time/Space Fractional Models

Fractional differential equations have been utilized as powerful mathematical tools for accurate description of challenging phenomena.In most cases,fractional differential equations are more complicated than the integer analogs and cannot be solved analytically,so numerical methods have been an efficient means for the modeling of fractional differential equations.However,due to the nonlocal nature of fractional differential operators,numerical discretizations of space-fractional differential equations usually yield dense stiffness matrices.This is deemed computationally very expensive,especially for multi-dimensional problems.Based on previous studies on fast algorithm and ADI finite difference method,this paper proposes a fast ADI difference algorithm with high dimension and high accuracy,which is applied to two different fractional diffusion models,time and space,and extended to nonlinear cases.Variable-order time-fractional diffusion equations,which can be used to model solute transport in heterogeneous porous media are considered in the third chapter.Based on the well-posedness and regularity theory^[59],the multidimensional variable order efficient ADI finite difference scheme is proposed.At the same time,Thomas algorithm is used to reduce calculation and storage,and theoretical proof is strictly carried out.In addition,ADI method for unconditionally stable three-dimensional time fractional diffusion equation with variable order is extended.Finally,several numerical examples are given to validate the theoretical analysis and show efficiency of the ADI methods.In the fourth chapter,a three-dimensional time-dependent Riesz space-fractional diffusion equation is considered,the corresponding difference scheme is given,a high precision and fast ADI difference scheme is proposed.The method is proved to be unconditionally stable and convergent with second-order accuracy both in time and space with respect to a weighted discrete energy norm.Efficient implementation of the method is carefully discussed,and then based on fast matrix-vector multiplications,a fast conjugate gradient（FCG）solver for the resulting symmetrical linear algebraic system is developed.Numerical experiments support the theoretical analysis and show strong effectiveness and efficiency of the method for large-scale modeling and simulations.In chapter fifth,we based on the linear model in chapter 4,we explore the three-dimensional nonlinear Riesz space fractional diffusion equation,and propose ADI schemes based on second-order-extrapolation linearized method and Newton linearization,and verify the effectiveness of these two schemes.