The High-order Numerical Algorithms For The Space And Time Fractional Bloch-Torrey Equation

Recently, fractional differential equations have attracted increasing interests mainly because they have been proved to be successful in modeling many phenomena in engineer-ing, physics, chemistry, biology and even economics. Due to the intrinsic nonlocal property of the fractional derivatives, the fractional differential equations have been proved to be promising to describe some phenomena or processes with memory and hereditary. These important applications have led to an intensive effort recently to find accurate and stable numerical methods that are also straightforward to implement. This paper is devoted to constructing high-accuracy numerical methods for the time and space fractional Bloch-Torrey equation with finite difference scheme, and the corresponding error estimates are given.The article is divided into two parts.In the first part, we propose two high order difference schemes for one-dimensional time and space fractional Bloch-Torrey equation and give the corresponding priori esti-mates. Firstly, a third-order accurate formula based on the weighted and shifted Grunwald-Letnikov difference operators is used to approximate Caputo fractional derivative in tem-poral direction. For the discretization of spatial fractional derivative, we discretize Riesz derivative by fractional center difference operator and get a formula with a second-order accuracy to approximate Riesz fractional derivative. Thus a new high order difference scheme A is derived for one-dimensional time and space fractional Bloch-Torrey equation, with truncation errors of order 3 in time and of 2 in space respectively in L1(L2)-norm. The unique solvability, unconditional stability and convergence of the scheme are rig-orously proved by the discrete energy method. A numerical example is given to verify the accuracy of numerical solution and efficiency of the difference scheme. Secondly, for the discretization of the Caputo fractional derivative in temporal direction, we still use the method just mentioned; For the discretization of spatial fractional derivative, we use the fractional center difference operator to approximate the weighed values of the Riesz derivative at three points and get a fourth-order approximation in the spatial direction. So another new high order difference scheme B is established with truncation errors of or-der 3 in time and of 4 in space respectively in L1(L2)-norm. Likewise, we use the discrete energy method to prove the unique solvability, unconditional stability and convergence of the scheme respectively and give a numerical example to verify the accuracy of numerical solution and efficiency of the difference scheme.In the second part, we establish two high order difference schemes for two-dimensional time and space fractional Bloch-Torrey equation and give the corresponding priori esti-mates; Then four ADI schemes are given for two-dimensional time and space fractional Bloch-Torrey equation. Firstly, for the discretization of the Caputo fractional derivative in temporal direction and Riesz fractional derivative in spatial direction, we adopt the same method as used in constructing the difference scheme A for the one-dimensional time and space fractional Bloch-Torrey equation. A third-order accuracy formula is used to approximate the Caputo fractional derivative. A second-order accuracy formula is used to approximate the Riesz fractional derivative. Consequently, we construct a new high order difference scheme C for two-dimensional time and space fractional Bloch-Torrey equation, with truncation errors of order 3 in time and of 2 in space respectively in L1(L2)-norm. The unique solvability, unconditional stability and convergence of the scheme C are rig-orously proved by the discrete energy method. A numerical example is given to verify the accuracy of numerical solution and efficiency of the difference scheme C. Secondly, for the spatial fractional derivative, we utilize the same method as used in constructing the difference scheme B for the one-dimensional time and space fractional Bloch-Torrey equa-tion and then achieve a fourth-order accuracy approximation in spatial direction. Hence, we get a higher order difference scheme D which is temporal third-order accuracy and spatial fourth-order accuracy in L1(L2)-norm respectively. Similarly, we use the discrete energy method to prove the unique solvability, unconditional stability and convergence of the difference scheme D and give a numerical example to verify the accuracy of nu- merical solution and efficiency of the difference scheme D. Finally, we establish four ADI schemes for the two-dimensional space-time fractional Bloch-Torrey equation. Although ADI scheme can simplify the problem, it only has α-order or 2a-order accuracy, which is lower than the accuracy of difference schemes C or D. Therefore, we only give the ADI schemes and have no further research in theory.