The Finite Difference Schemes For Solving Two Kinds Of Partial Differential Equations

In the ?rst part, two implicit ?nite di?erence schemes are developed for solving one dimensional space-time fractional convection-di?usion equations with variable coe?cients on a ?nite domain, this paper’s purpose is to compare these two ?nite di?erence schemes in terms of the accuracy and convergence order of the scheme. It takes advantage of central di?erence at two di?erent point respectively, when the paper discretizes the temporal α-order derivative term using Caputo fractional order derivative. The spatial β-order derivative term is discretized with using a shifted Gr¨unwald formula. According to the analysis of stability and convergence of the methods done in the paper, it can draw the conclusion that the ?rst scheme more e?ective than the second one. Numerical examples with known exact solution are used to verify and compare the?nite di?erence schemes, it also illustrates the consistency of the theory and the experiment.In the second part, a high-order exponential di?erence scheme is put forward for solving the problem of one- and two-dimensional wave equations, the paper?rstly uses central di?erence quotient to discrete the spatial second derivative term and obtain the system of semi-discreted di?erential equations. By solving the equations with the initial vector and discreting the time variable, a iterative format is obtained, iterative coe?cient of which has a exponential form,then the iterative coe?cient is modi?ed using Trotter Product formula of the exponential function. The modi?ed exponential form is approximated by the(2,2) Pad′e approximation. A high-order exponential di?erence scheme of temporal second-order accuracy and spatial fourth-order accuracy is developed for solving one- and two-dimensional wave equations, this method has the characteristics of a small amount of calculation and high precision. The paper has also proved unconditional stability and convergence of the exponential di?erence scheme. Finally the numerical experiment is conducted to verify the accuracy and availability of the method.