Research About Numerical Solutions Of Low Summed Partial Differential Equations

As the growing demand for energy, the energy consumption is growing. The scarcity of resources becomes one problem that people are care about. The researches about the storage state of underground resources is important. It has been widely applied especially in numerical reservoir simulation. The research of solution of fluid mechanics in porous equation becomes an important issue. For complex partial differential equations called low summed partial differential equation and its analytic solution is unable to solve. Usually we solve them by simplified but the error of simplified can’t be avoided. So the paper uses numerical method to solve partial differential equation. It includes finite difference method, finite element method and so on. This paper mainly introduce the traditional finite difference method and its improving method. In order to obtain higher precision, stability and easily solved scheme, this paper has calculated and proved. There are a variety of difference scheme. Two-tier scheme such as:explicit difference scheme, implicit difference scheme and Crank-Nicolson scheme. Three-tier scheme such as:Richardson difference scheme and Dufort-Frankel difference scheme. On the basis of traditional difference scheme, this paper presents three-weighted implicit scheme. It has been proved that the improved scheme has higher precision, is unconditionally stable, its format is simple and is easy to solve. For the difference equation that has two-dimensional in space, it must first select the appropriate sort method. This paper compared standard arrangement format, diagonally format, point alternately format and alternating diagonal format. It has proved that the alternating diagonal format can save the storage and improve the computational efficiency. The final step is to solve the equations. The elimination methos is commonly used but not applicable to solve large sparse equations. Forward elimination and backward substitution method and iterative method have advantages in solving large sparse equations. The advantages are occupy less computer memory, simply calculation and so on.