Research Of High-accuracy Compact Difference Scheme For Solving Convection-Diffusion Equation Based On Exponential Transformation

The convection-diffusion equation is one of the very important equations of motion in computational fluid dynamics,it can describe many physical phenomena,for example,the diffusion of pollutants in rivers and atmosphere,fluid flow,and heat conduction.but the actual application problems of convection-diffusion equations are often more perplexed,it is not easy to get solutions,thus numerical solution methods are very significant.So,this work bases on the idea of finite difference to construct three high-order compact difference methods of the unsteady convection-diffusion equations.In this paper,we study the following aspects:(1)An exponential transformation method for constructing high-order compact schemes to solve unsteady convection-diffusion equations,to eliminate the difficulty in dealing with convection terms.A high-accurate compact difference scheme is constructed for solving the one dimensional diffusion equation by the Pade' formula combined with spline interpolation formulas.The numerical experiments prove the accuracy and reliability of present method.(2)Based on the high-accuracy compact finite difference scheme for one dimensional convection-diffusion equation,a new high-order compact finite difference scheme is deduced for two dimensional unsteady convection-diffusion equation.The reliability of the scheme is demonstrated by numerical examples.(3)An exponential transformation method for constructing high-order compact schemes to solve unsteady convection-diffusion equations,to eliminate the difficulty in dealing with convection terms.A high-accurate compact difference scheme is constructed for solving the one dimensional diffusion equation by the fourth-order compact difference formula combined with extension of the 1/3-Simpson formulas.The numerical experiments prove the accuracy and reliability of present method.