Research And Application Of High-resolution Numerical Methods For Solving Convection-Diffusion Equations

The convection-diffusion equations are used to show a lot of physical phenomena,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 volume to construct two methods of the convection-diffusion equations,which poss high resolution,inherent conservation of the equation and high order accuracy within small stencils.In this paper,we study the following aspects :First,the vast majority of the flow and heat transfer problems of mathematical formula are introduced, discrete finite volume method is described of that kind of equations.A numerical of examples are presented to compare the different discrete schemes about the stability and convergence precision.Secondly,a new modified high-resolution combination scheme is developed based on the definition of Leonard normalized variable,and the sufficient condition for convergence of these combination schemes are deduced.Numerical examples explained that the new scheme not only has high resolution,and can capture large gradients of physical changes. As a result,the numerical result of the new scheme is better than the traditional combination ones.Third,the high-order Padé compact finite volume scheme is proposed by the exponential transformation and the Padé approximation compact finite volume scheme and is stable.Finally,some numerical examples are presented to verify the validity of the new scheme,as well as the advantages of high accuracy and high resolution in dealing with the boundary layer problems or locally large gradient problems.Finally,the high-resolution schemes are applied to solve fluid mechanics problems,the parallel pipe and the water quality model are exampled to verify the new schemes.