High-order Compact Difference Schemes And Mesh Adaptive Method For Diffusion-reaction Blowup Problems

Nonlinear diffusion reaction blowup question has very important applications in the fields of chemistry,biology,physics and engineering.Currently,the blowup of solutions of nonlinear equations has not only arouses the interest of many workers of partial differential equations,but also attracts extensive attention in the fields of quantum mechanics,fluid mechanics,nonlinear optics.In this thesis,the finite difference method and the mesh adaptive method for nonlinear diffusion reaction equations are studied.Firstly,the Crank-Nicolson scheme is used in the time direction and the truncation error remainder correction method is used in the space direction.a high-order compact difference scheme for one-dimensional nonlinear diffusion reaction equation is established on a non-uniform grid.And a high-precision scheme with fourth-order accuracy in space and second-order accuracy in time is derived.The stability of the scheme is analyzed by Fourier method.In solving the problem of blowup,since the exploding solution will suddenly become unbounded in limited time,we have established the adaptive algorithms of time and space grid respectively,which can encrypt the grid near the point of space explosion.A small time step is used in the vicinity of the time burst point.This method is extended to the two-dimensional problem,and the high-precision compact ADI difference scheme and the mesh adaptive method for the two-dimensional nonlinear diffusion reaction equation are established.Finally,the scheme of this paper is verified by the problem with exact solution.On the basis of this,some blow up problems without exact solution are simulated directly.The simulation results reveal the asymptotic behavior of numerical solution and the blow up phenomenon,and the initial conditions,critical size and critical time of the explosion phenomenon.The location of the space in which the explosion took place,etc.It can be concluded that the results obtained in this paper are in good agreement with those in the literature,and our results are proved to be accurate and effective.All the schemes and problems in this paper can be implemented in the software for solving partial differential equations.