The Study On Compact Finite Difference Methods For Several Types Of Evolution Equations

The thesis mainly concentrates on the developments of the compact finite difference methods applied to solve several types of evolution equations,and corresponding theoretical findings are obtained as well.Some numerical examples are provided to test the accuracy and efficiency of the proposed numerical algorithms.The thesis concludes five parts.The concrete research jobs are given as follows.The second chapter is devoted to the development and application of two high-order numerical methods for solving one-dimensional Burgers' equations,which are both fourth-order accurate in both time and space.One of them is based on the uses of Crank-Nicolson method combined with Richardson extrapolation method for temporal integration and fourth-order compact finite difference approximation for spatial discretization.Additionally,a combination of fourth-order time stepping method based on padé approximation for temporal discretization with fourth-order compact finite difference method for spatial approximation yields the other fourth-order method.Utilizing matrix analysis methods,we study their stability.Numerical experiments illustrate the feasibility and high efficiency of new algorithms.In chapter 3,first of all,a compact difference scheme is established for one-dimensional nonlinear reaction-diffusion equations with a fixed delay.By the energy method,it is proved that the difference solution converges to exact solution with a convergence order of O(t~4+h~4)in L~?-norm.Then,a Richardson extrapolation method is applied to make the final solution fourth-order accurate in both time and space.Besides,the extensions of the solver to other complex delay problems are studied in detail.Finally,numerical results demonstrate the computational accuracy and efficiency of our algorithms.In chapter 4,a three level compact finite difference scheme for solving a one-dimensional viscous wave equation is derived.By using the energy method,it is proved that the difference solution converges to exact solution with a convergence order of O(t~4+h~4)in the maximum norm.Moreover,the Richardson extrapolation method is utilized to make the final solution fourth-order accurate in both time and space.Finally,a numerical example is provided to verify the convergence order and the validity of the difference scheme.The last chapter is suggested for the conclusions and some promising future researches.