Extrapolations Of Finite Difference Methods For Two Kinds Of Parabolic Equation And Stability Analysis

Parabolic partial differential equations are a kind of important partial differential equation.it can be used to describe many physical phenomena,and are widely applied in science and engineering.Therefore,it is very important to study and construct a simple and efficient numerical algorithm.In this thesis,we discuss the extrapolation finite difference method of two kinds of parabolic equation,that is unsteady convection-diffusion equation and the heat equation.In the first part,we discussed the characteristic finite difference method for convection-diffusion equation.For convection-dominated problems,the convection-diffusion equation presents the hyperbolic character.The characteristic difference method is developed based on the combination of the characteristic line of hyperbolic equation and the difference method,and its fundamental advantage is that the time-truncation error is small.The traditional characteristic difference method has only one or two order for time.In this thesis,two order and three order extrapolation-characteristic finite difference schemes for the initial boundary value problems of the convection-diffusion equation are constructed by combining the characteristic difference method with the extrapolation method,thus time accuracy was improved,and the error estimation and stability analysis were carried out.Finally,a numerical example was given to illustrate the validity and reliability of scheImes.In the second part,we studied the spurious oscillations of Crank-Nicolson(C-N)extrapolation method for solving parabolic equations with discontinuous initial conditions(i.e.numerical dispersion effect).When the C-N method is used to solve the heat equation of the discontinuous initial condition,if the time step size k and the space step size h do not satisfy the condition that k/h<X/?,the numerical solution appears spurious oscillations.Similarly,the L0-stable C-N extrapolation algorithm is also oscillating when this kind of problem is solved.The property of finite difference method can not be described by the traditional properties of the finite difference method such as stability and convergence,but it involves the internal microscopic characteristics of the difference scheme.Therefore,it is very urgent to study the theoretical analysis method of the numerical dispersion of difference schemes.In this thesis,The relationship between the spurious oscillation and discontinuous initial condition and the growth factor is given,and a new constraint on the time step size and the space step size is given,and the results are generalized to the n dimensional case.