Fully Implicit Upwind Finite Difference Scheme And Iterative Method For Nonlinear Convection-diffusion Equations

In this paper,to realize high-efficiency and high-fidelity solution for nonlinear convectiondiffusion equations with Neumann boundary conditions,a fully implicit upwind finite difference(FIUFD)discrete scheme and its nonlinear iterative method are designed,and their fundamental properties are analyzed.The study is carried out for one-dimensional convection-dominated model problem on non-uniform grids.A second-order time accuracy FIUFD discrete scheme is constructed to avoid non-physical numerical oscillations and dispersions and simulate vividly the problem with transient physical quantity.A Picard iterative method matching well to the nonlinear upwind difference scheme is designed to solve the nonlinear problem efficiently.Through careful derivation,the equivalent expression of the error equation between the solution of the FIUFD scheme and the exact solution of the equation is obtained.By using energy estimate and related argument techniques,theoretical analysis is performed.Wherein,a new reasoning method is developed to overcome the difficulties on the nonlinear diffusion flux brought by Neumann boundary conditions and the difficulties on the nonlinear convection flux brought by the upwind discretization.Consequently,the theoretical analysis on the fundamental properties of the FIUFD discrete scheme and the iterative method is successfully performed.It is proved that the solution of the nonlinear FIUFD scheme exists,is unconditionally stable,and has the second-order temporal and first-order spatial convergence accuracy to the solution of the original problem.While the Picard iterative method has the same convergence accuracy.Numerical experiments on non-uniform grids are carried out,and compared with the first-order time accuracy FIUFD scheme and the traditional fully implicit central finite difference scheme.It is verified that the proposed scheme permits larger time step-lengths,has high accuracy and computational efficiency,and can avoid numerical oscillation.The ideas and methods in this paper can be extended and developed to multi-dimensional problems,to FIUFD schemes with first-order time accuracy or second-order spatial accuracy,as well as schemes with finite volume and other spatial discretizations.