Difference scheme of the convection-diffusion equation
|School||University of Electronic Science and Technology|
|Keywords||convection-diffusion equation finite difference scheme exponential transformation exponential difference scheme stability|
Convection-diffusion equations are fundamental equations for dynamics, and are the linearized model of the nonlinear equations coming from the viscous fluid dynamics. They are of their own value in describing many natural phenomena ,such as, the diffusi- on of polluted substances in water , air, the diffusion of heat and salinity in the ocean, and so on.Many problems about fluid dynamics are appealed to the convection-diffusion equations. So, the computation of the convection-diffusion equations has very important significance in theory and practice.There are many numerical methods for convection-diffusion,for example,finite dif- ference method, finite volume method, finite elements method, etc.The finite difference method is one important numerical method for fluid dynamics.It has been developed for more than eighty years, got great success. Especially, there are many achievements in recent twenty years, von Neumann, Courant, Friedrichs, Lax, Wendroff and many other mathematics had done a lot of hard work.In this paper, the exponential difference methods for convection-diffusion are proposed. For one-dimensional convection-diffusion equation, at first, the differential equations is semi-discrete approximation. Then, from the exponential transformation, the diffusion term is eliminated. Structured difference schemes for the equation after exponential transformation, the exponential difference scheme is got using the inverse exponential transformation. For the two-dimensional convection-diffusion equation, we get higher accuracy scheme of the Poisson equation using operator method firstly. Secondly, with the exponential transformation the one-order diffusion term is eliminated. The exponential difference scheme is structured combining inverse exponential transformation and the Poisson equation’s higher order difference scheme.With theoretic analysis, we know that these exponential difference schemes have higher accuracy and good stability. Numerical tests prove those schemes’effective.