Two-parameter Optimal Circulant Preconditioning And Spectral Analysis Of Riesz Fractional Advection-dispersion Equation

For the discretized linear system of the Riesz fractional convection-diffusion equation,this paper studies the generating function of the coefficient matrix of the Toeplitz structure in the linear system,and uses the theory of the generating function to analyze eigenvalue range of such Toeplitz matrices.On this basis,we establishes a simplified proof of the convergence of the two-parameter splitting iterative method,and constructs a two-parameter splitting optimal circulant preconditioner through the iterative method.Theoretical and numerical results show that the eigenvalues of the preconditioned matrix are clustered and are all clustered around 1,which can well speed up the convergence of the Krylov subspace iteration method.Numerical results also show that the two-parameter splitting optimal circulant(TPST)preconditioner is significantly better than the two-parameter splitting Strang circulant(TPSC)preconditioner.