Sixth-Order Finite Difference Schemes For Solving Nonlinear Convection Diffusion Reaction Equations

The unsteady convection diffusion reaction equation has been widely concerned by researchers because of its very important practical application value.In this thesis,we discuss the high-order accuracy finite difference method of the equation model,construct the spatiotemporal sixth-order difference scheme,and verify the accuracy and reliability of the presented schemes by numerical experiments.Firstly,a five-point sixth-order difference operator is derived for the second derivative.The space terms of the one-dimensional quasilinear diffusion reaction equation are discretized by this operator,and the adjacent boundary points are calculated by means of two virtual points.The time derivative is discretized by the Crank-Nicolson method,and the time accuracy of the scheme is improved from the second-order to the sixth-order by the Richardson extrapolation method.A two-layer implicit difference scheme with the sixth-order accuracy is established.The linear system formed by the difference scheme is solved by the penta-diagonal Thomas method,and the nonlinear source term and the two virtual points are solved by the iterative method.By Fourier method,the unconditional stability of the scheme is obtained.Numerical experiments verify the accuracy of the scheme and the feasibility of using the virtual points’method to deal with the adjacent boundary points.Then,for the generalized Burgers-Huxley equation and the generalized Burgers-Fisher equation.The internal points of the space are discretized by five-point difference operator and truncation error correction technique to obtain a five-point sixth-order scheme,and the adjacent boundary points are discretized by the same idea to obtain a three-point fourth-order scheme.The time term is discretized by the sixth-order backward difference formula,the Crank-Nicolson scheme combined with the Richardson extrapolation method is used as the starting layer scheme.A seven-layer sixth-order implicit difference scheme is established.The linear system formed by the difference scheme is solved by the penta-diagonal Thomas method,and the nonlinear part is solved by the iterative method.Finally,the accuracy and reliability of the proposed scheme are verified by numerical experiments,and the feasibility of using lower order small template scheme to deal with adjacent boundary points is also verified.Finally,the difference methods for solving the one-dimensional generalized Burgers-Huxley equation and the generalized Burgers-Fisher equation were extended to establish a seven-layer sixth-order implicit difference scheme for solving the two-dimensional nonlinear convection diffusion reaction equation.The Gauss-Seidel iterative method is used to solve the linear system formed by the difference scheme.Finally,numerical experiments show that the scheme has sixth-order accuracy in calculating coupled and multicoupled equations with different boundary conditions.