Research On High Precision Difference Algorithm For High - Dimensional Sine - Gordon Equation

The numerical calculation of partial differential equations is an indispensable tool in natural science, technical science, engineering science and other important fields. For example, the issues often involve all kinds of partial differential equations of high dimension in meteorology, oil exploration, aerospace technology and other fields. Hence the solution of these partial differential equations has become the core contents in science and engineering calculation. The high dimensional sine-Gordon equation is a kind of nonlinear hyperbolic partial differential equation, which is widely used in various fields. With the development of science and technology and high performance computer, the method for high dimensional sine-Gordon equation, which owns unconditional stability, small size computation and high precision, has become an urgent matter.High accuracy finite difference method is one of the most common and effective numerical method for solving partial differential equations. This thesis studies high accuracy finite difference method for two-and three-dimensional sine-Gordon equations. First of all, by combining the compact difference scheme and alternating direction implicit scheme, we obtain alternating direction implicit compact difference scheme of three layers. The compact difference scheme can use less computation nodes to achieve high precision, on the other hand, alternating direction implicit scheme can convert the high dimensional problem into a series of one-dimensional problem, we just have to use Thomas’ method to solve the linear algebraic equations of the coefficient of three diagonal matrix. Secondly, it is shown by the energy method that compact alternating direction implicit scheme has second-order temporal accuracy and fourth-order spatial accuracy. Finally, an implemental Richardson extrapolation method is developed to improve temporal accuracy from second-order to fourth-order. The finite difference scheme has four-order accuracy. The results of the numerical examples show that the alternating direction compact scheme and improved extrapolation method can reach the accuracy of scheme respectively.