Research On Preconditioning Technology Of Fractional Diffusion Equation

As a very important part of differential equations,fractional diffusion equations have attracted much attention in mechanics and physics.In addition,it is also widely used in many fields such as chemistry,biology,and image processing.Because the fractional order is often able to describe the phenomena and results more accurately than the integer order differential model,it is also very important to solve this type of equation.For this type of equation,on the one hand,because the analytical solution that can be used as a closed form is usually not available,numerical methods have become the main method for solving its approximate solutions.On the other hand,due to the non-local nature of fractional operators,the use of simple discretization,even if it is implicit,will lead to unconditional instability,and most FDE numerical methods tend to generate full coefficient matrices,which leads to the need for storage of O(n~2)and computational complexity of O(n~3)when the direct calculation method is used to solve the problem.Here n represents the number of spatial grids.In order to save costs and speed up calculations,finding an efficient and fast numerical solution has become the focus of many scholars.Among them,the method of accelerating the solution of fractional differential equa-tions by constructing the preconditioner of the coefficient matrix has achieved good re-sults and has become a powerful tool for solving such equations.And through the efforts of many researchers,many classic and effective preconditioners have been constructed.This article also starts from the construction of an efficient and fast numerical solution method,by introducing the matrix splitting band substitution method of linear equations,constructing the preconditioner of the coefficient matrix,thus accelerating the solution of the Krylov subspace iteration method(1)Diffusion from the spatial fractional order The diagonal plus Toeplitz linear system discretized in the equation?(2)Starting from the time-space fractional diffusion equation,discretizing through finite differences in the space-time framework,a fully coupled discrete linear Kronecker product-sum linear system is obtained.For the linear system generated in(1),this paper proposes a two-parameter two-step split iterative GDTS method to deal with?for the linear system generated in(2),a single-parameter two-step split iterative TTS method is proposed.According to these two methods,the iterative matrix is obtained separately,and through the iterative matrix spectral radius is less than 1,the iterative method converges this condition,establishes the convergence theory,derives the asymptotic convergence upper bound,and the optimal value of the iterative parameter.Finally,the Strang's circulant matrix is introduced to replace the Toeplitz part of the iterative matrix,thereby constructing the preconditioner of the coefficient matrix.Since the Toeplitz matrix is replaced by a circulant matrix in the preprocessor,the calculation amount of matrix vector multiplication will also be reduced to O(n log n).It is proved by theoretical derivation that the coefficient matrix can be expressed as the sum of an identity matrix,a low-rank matrix and a small norm matrix after the pre-processing sub-processing constructed in the article.Numerical experiments also show that the preconditioner con-structed in the article can effectively concentrate the spectral radius of the original coef-ficient matrix,and when the Krylov subspace iteration method,such as gmres,is used to solve the problem,the number of iteration steps is significantly reduced,which speeds up the solution process.