Sparse Grid Method For Space Fractional Partial Differential Equations

Space fractional derivative equations can be used to simulate many phenomena,but it is still difficult to calculate the numerical solutions and corresponding analysis of space fractional derivative partial differential equations,especially for high-dimensional space fractional derivative problems.The geometric growth of computation caused by dimension growth also puzzles fractional equation.The sparse grid method is used to solve the problem of dimension curse.By using tensor basis and removing the nodes that have less impact on error,the sparse grid is formed and the computing cost is reduced.In this paper,we mainly consider the sparse grid algorithm for R-L type fractional diffusion equation and tempered R-L space fractional diffusion equation.In chapter 1,we introduce the development history of fractional derivative and sparse grid method,and the work of others.Then we introduce some basic knowledge of fractional derivative.In chapter 2,we introduce some knowledge of sparse gridand the projection trans-formation between the difference operator in full grid and the difference operator in sparse grid.In Chapter 3,we discuss the sparse grid method for the fractional diffusion equation in R-L space,give the truncation error of the difference operator on the sparse grid,and analyze the convergence and stability of the difference scheme.Compared with the full grid method,it is found that the computational cost of the sparse grid method is much less than that of the full grid method,and the cost of the sparse grid method is much less when the dimension is increased.In chapter 4,the sparse grid method on the R-L fractional order equation is ex-tended to the tempered R-L fractional equation,and the corresponding truncation error,stability and convergence analysis are given.In chapter 5,the paper summarized and some questions were discussed.