Two Numerical Methods For Parabolic Equation

The contents of this dissertation are divided into three chapters.In Chapter One,a new method of approximating the solution of second-order parabolic system using reproducing kernel function is devised. The time discretization are formulated by Laplace-modified Galerkin procedure, and the approximating solution at cach discrete time stop is given with explicit formula. The stability and error estimate are derived by using the energy method. Some numerical results arc presented.In Chapter Two, apply the reproducing kernel function method to one-dimensinal nonlinear convection-diffusion problem. The stability and error estimate are proved with energy method. Some numerical results are presented.In Chapter Three, the characteristic local discontinuous Galerkin method for solving nonlinear Burgers equation is devised.In this method, a characteristic procedure approximates nonlinear convection term and a local discontinuous Galerkin method approximates diffusion term. The method keeps the advantage of characteristic method, can be calculated with large time step, also it brings numerical flux into finite element method. An optimal error estimate is derived with energy method.