Preconditioned Methods For The Third Order Radau-?A Discretization Of One-dimensional Space Fractional Diffusion Equations

Fractional diffusion equations(FDEs)are currently considered to be one of the most commonly used models for describing anomalous diffusion.They have been proposed and studied in many research fields.Generally,most FDEs cannot obtain analytical solutions.With the increase of applications using FDEs and the high requirements of analytical solutions,numerical methods for FDEs have attracted much attention and have developed rapidly.Radau-IIA method is a kind of multi-level high-order implicit Runge-Kutta method.It has high accuracy and high stability.When dealing with stiff problems,its stability is better than the explicit Runge-Kutta method.As we all know,the non-locality of fractional differential operators makes the numerical solution of FDEs usually require a huge amount of calculation.For the linear system obtained after discrete,if we directly calculate,not only is the calculation complicated,but the convergence rate is slow.The purpose of this paper is to construct a fast preconditioned iterative methods for the three-order Radau-IIA discrete system of one-dimensional space fractional diffusion equations.High-order method is used to discretize the time of fractional diffusion equation,and an effective preconditioned method is constructed for the resulting coupling system.The accelerated GMRES iteration method is used to speed up the iteration and improve the computational efficiency.The main content of this article is as follows:The first part,we study the construction of an effective preconditioner for the system generated by the implicit Runge-Kutta time discretization method of one-dimensional space fractional diffusion equation.We consider the third-order,two-stage Radau-IIA method,which results in a coupled 2×2 block system.Our method is based on a Schur complement formulation for the unknown at the second stage.We present an approximate factorization of the Schur complement and derive the eigenvalue bounds of the preconditioned system.We observe that the components of the approximate factorization have the same structure as the system derived from implicitEuler discretization of the problem.Therefore,we reuse the high-performance implicit Euler discrete preconditioners as the building block of our preconditioners.Several numerical experiments are presented to show the effectiveness of our approaches.In the second part,we study the construction of an effective preconditioner for the system generated by the implicit Runge-Kutta time discretization method of one-dimensional nonlinear space fractional diffusion equation.For time discretization we maintain the third-order two-stage Radau-IIA method.A preconditioner is constructed based on a matrix equation with a Kronecker product form a nonlinear term in the discrete system.Our method is to use the fixed-point iterative method to optimize the nonlinear term,and use the structure-preserve method to approximate the large dense matrix generated after discretization.Several numerical experiments are presented to show the effectiveness of our approaches.