Three Kinds Of Numerical Methods For The Convection-dominated Diusion Problems

The mathematical model of the physical processes accompanied by material transportand diffusion and the ffow of viscous ffuid are convection-diffusion equation or the partialdifferential equations containing such equation. The numerical simulation of convection-dominated diffusion problems is important in theoretical and practical, and can be usedfor energy exploitation, environmental science, ffuid mechanics, electronics, and so on.It is well known that the convection-dominated diffusion problem has strongly hyper-bolic characteristics, the stand methods, such as finite difference or finite element methodfor the convection-dominated diffusion problems will produce excessive numerical diffu-sion or nonphysical oscillations. In 1982, Douglas and Russell presented the characteristicfinite difference method and characteristic finite element method for solving such prob-lems. The characteristics has proved effective in treating convection-dominated diffusionproblems. Error estimates and numerical experiments have shown that this method per-mits the use of large time steps, and avoids or sharply reduces the numerical diffusionand nonphysical oscillations.The finite volume method (FVM) (or Generalized difference method or Box method)is used to ffuid and underground ffuid computations. This method is based on two spaces:the trial space of piecewise polynomial functions over the primal partition and the testspace of piecewise constant functions over the dual partition. The FVM methods not onlykeep the computational simplicity of the difference methods, but also enjoy the accuracyof the finite element method. Furthermore, they maintain the mass conversation law.The streamline diffusion finite element method (SD) for solving convection-dominateddiffusion problem is a highly effcient finite element method, which has a good numericalstability and higher precision. It used to solve time-dependent convection-diffusion equa-tion is based on time-space finite element space. Although time-space finite element spacecan well coordinate the space and the time ffow field, and it is easy for theoretical analysis,but it paid a high price (the enormous amount of computation and storage space), es-pecially for high-dimensional problems. The finite difference streamline diffusion method(FDSD) is proposed by Sun and his coworkers, that is, using SD method discrete only inspace variables and using finite difference discrete in time variables. Comparing with SD method the FDSD method not only simplifies the computational work but also keeps thegood stability and high accuracy.In this paper, we develop the characteristics method, AGE method, two-grid algo-rithm and FDSD method based on the work of the predecessor and give three kinds of ef-fective numerical methods for solving convection-dominated diffusion problems. The firstkind is characteristic-AGE method, the characteristic-difference scheme is presented firstlyfor the convection-dominated diffusion problems by the bilinear interpolation method.Secondly, the characteristic-AGE method based on the group explicit forms is proposedand the stability analysis is derived for the convection-dominated diffusion problems. Fi-nally, numerical experiments show that this method is prise, non-oscillatory and reducethe numerical diffusion. The second kind is Two-grid method for characteristics finite vol-ume element of the nonlinear convection-dominated diffusion problems, A characteristicsfinite volume element discretization technique based on two subspaces is presented for anonlinear convection-dominated diffusion problem. The solution of a nonlinear system onthe fine space is composed of solving one small nonlinear system on the coarse space and alinear system on the fine space. We present the two-grid method for characteristics finitevolume element of nonlinear convection-dominated diffusion problems and convergence es-timates are derived to justify the effciency of our methods. It is shown both theoreticallyand numerically, that the new scheme is effcient to the nonlinear convection-dominateddiffusion equations. The third kind is the characteristic-finite difference streamline dif-fusion method for two dimensional linear convection-dominated diffusion problems, Themethod of characteristics is combined with finite difference streamline diffusion (FDSD)method to create the characteristic FDSD (C-FDSD) procedures for two-dimensionalconvection-dominated diffusion equation, the analysis of stability and error estimated arepresented. In the end, numerical examples are presented to clarify the method. Thisscheme not only realizes the purpose of lowering the error of time, using large time stepof solving the two-dimensional linear convection-dominated diffusion equation, but alsokeeps the favorable stability and high precision of the FDSD method.