Time Two-grid Algorithm Based On Finite Difference Method For The Two Dimensional Nonlinear Volterra Equation With A Weakly Singular Kernel

In this thesis,a time two-grid finite difference(FD)algorithm is proposed for solving the two-dimensional nonlinear Volterra equation with a weakly singular kernel.This algorithm can be com-pleted by three steps.Firstly,a nonlinear FD system is solved on the time coarse grid of size ?C.Then,the Lagrange's linear interpo-lation formula is applied to provide some useful values for the time fine grid of size ?F.Finally,a linear system is solved on the time fine grid.In the temporal direction,a backward Euler method is employed for the time derivative and a first order convolution quadrature rule is applied to discretize the integral term.And the second order central difference quotient is considered for the spatial approximation.By means of the discrete energy method,we obtain the unconditional discrete L2 stability and convergence oforder O(?C2+?F+hx2+hy2).where hx and hy are the spatial step sizes in the x direction and the y direction,respectively.Additionally,numerical examples are presented to show the feasibility and efficiency of the time two-grid FD algorithm.The structure of this thesis is as follows:In Chapter 1 we in-troduce the background of the two-dimensional nonlinear Volterra equation with a weakly singular kernel,the development of the two-gird algorithm and the main contents of this study.In Chapter 2,we show some notations,related definitions and auxiliary lemmas.The time two-grid FD scheme for the two-dimensional nonlinear Volterra equation with a weakly singular kernel is established in Chapter 3.In Chapter 4,the stability and convergence of the time two-grid FD scheme are carried out.We also present three numerical examples,and compare the time two-grid FD method with the standard nonlinear implicit FD method in Chapter 5.Finally,a conclusion and the future work are given in Chapter 6.