A Crank--Nicolson--Type Compact Difference Method On The Uniform Time Mesh For A Class Of Weakly Singular Parabolic Integro--differential Equations

This thesis is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels.Due to the presence of the weakly singular kernel,the exact solution has singularity near the initial time t = 0.A Crank-Nicolson compact difference method is proposed for one/two-dimensional equations by designing a product integration rule for the integral term.In this method,time discretization is handled by a generalized Crank-Nicolson-type scheme and a compact difference approximation is used for the space discretization.The proposed method is constructed on the uniform time mesh,but it can still achieve the second-order convergence in time for weakly singular solutions.The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained.The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space.In order to improve the computational efficiency,the proposed method is combined with alternating direction implicit(ADI)scheme to construct a Crank-Nicolson-type ADI compact difference method for solving two/three-dimensional equations.The unconditional stability and convergence of the Crank-Nicolson-type ADI compact difference method are rigorously proved by using the discrete energy method.The optimal error estimate is obtained in the discrete L2-norm.The error estimate shows that the Crank-Nicolson-type ADI compact difference method has second-order convergence in time and fourth-order convergence in space.The theoretical and numerical comparisons between the proposed method in this paper and several known methods are also discussed.Numerical results confirm the theoretical analysis results and show the effectiveness of the proposed methods.