Two Level Finite Difference Method For Burgers Equation

Burgers equation is a kind of very important dynamic model. This equa-tion has wide application in many fields, such as gas dynamics, elasticity, waterresources pollution, as well as continuous random processes. The definite solu-tions of this problem often associated with shock wave phenomena, which bringgreat difculty to solve numerical solutions. Therefore, using efcient numericalmethod to solve Burgers equation has great theoretical and practical significance.Burgers equation was given in the study of fluid motion by Bateman in1915. Then, Lagerstrom and others in the study of one-dimensional case foundthat it is approximation model of unsteady Navier-Stokes equation. It does notcontain pressure gradient, but incorporates both the nonlinear convection anddifusion term, which is the most original describe mutual influence betweenthe convective difusion model, a good model for the numerical solution of thecomplicated N-S equations. Burgers equation can not only as a simplified modelof N-S equations in fluid dynamics, but also in the shallow water wave problemas well as contemporary mathematical model of trafc flow dynamics.In this paper, a two-level finite diference scheme is presented for the numer-ical approximation of Burgers equation. The full nonlinear problem is solved ona coarse grid with space size for H, and a linear problem is solved on a fine meshwith mesh size for h. This is an implicit method with unconditional stability.This method we study provides an approximate solution with nearly the sameerror as the usual one-level solution, due to the fine mesh is needed in the singlelevel methods to solve a large nonlinear problem. Thus it will cost too much computing time. Our proposed method does not need to solution of nonlinearequations. Hence, our method can save a large amount of computational time.Meanwhile, in this article also discusses Block centered finite diference methodfor Elliptic Problems in a rectangular region of a class of with the Second Bound-ary Value condition. Which combines the finite diference method for simple andadvantages of mixed finite element method with high precision. In one dimen-sional case, we discuss the two-point boundary value problems on uniform gridwith discontinuous coefcients and numerical experiments are presented; In thetwo-dimensional case, the block central finite diference method is discussed fornon-uniform grid and uniform grid, respectively. Numerical experiment resultsshow that, non-uniform grid problem is efect better for large gradient.This work consists four sections. Section1is preface. We introduce researchbackground, purpose and significance, and describe the research situation ofnumerical solution for Burgers equation. Finally, the organizational structure ofthis work is given.In section2, linearized Crank-Nicolson finite-diference method is presentedto numerical solution of two-dimensional Burgers equation. The proposed schemeis unconditionally stable, and second-order accurate in both space and time.Since this scheme is nonlinear, so, we need to change the nonlinearity item uinto linear item. Numerical experiments show the high accuracy and efciencyof the proposed diference scheme.In section3, two level finite diference method for Burgers equation ispresented, and it is unconditionally stabile of implicit diference scheme. Themethod we study provide an approximate solution with nearly the same error as the usual one-level solution. In this paper we prove the stability of the method,numerical results are in line with theoretical analysis of this problem. It is showedthat the scheme is efective.In section4, we give the Block centered finite diference method of ellip-tic problems. By introducing a derivative variable u constructing discrete blockcentral diference method. One dimensional case, two-point boundary value prob-lems with discontinuous coefcients are discussed in uniform grid. In two dimen-sional case, in non-uniform grid and uniform grid, block central finite diferencemethod is discussed respectively. Meanwhile, several numerical experiments aregiven to verify the numerical results.Section5is conclusion, we make a conclusion on the whole work. Numericalcomparison is carried out for the numerical method, and it is found that thenew format is implicit unconditionally stable. The method we study provides anapproximate solution with nearly the same error as the usual one-level solution.Our scheme can save a lot of CPU time. In this article it also gives the Blockcentered finite diference method of elliptic problems. Numerical results showthat, nonuniform grid problem is efect better for large gradient.