The Positivity-Preserving Scheme For Convection-Diffusion Equation And Numerical Methods For Equilibrium Radiation Diffusion Equation

The content of this thesis concludes three parts:(1)Construction of positivity-preserving finite volume scheme for a unsteady convection-diffusion equation and the proof of the exis-tence of its solution;(2)Numerical analysis of the fully implicit finite difference scheme for a diffusion problem with time-derivative term of conservative nonlinear energy and nonlinear iteration methods for equilibrium radiation diffusion equation;(3)Analysis of fully implicit fi-nite volume scheme for nonlinear diffusion problem and its application to equilibrium radiation diffusion equation based on Saha ionization model.In the first part,a nonlinear positivity-preserving finite volume scheme is developed for unsteady advection-diffusion equation on star-shaped polygonal meshes.By integrating the techniques of introducing associated unknowns at grid edges,smoothing the nonlinear coeffi-cient in discrete flux,and correcting the advection operator with modification,we design a high-fidelity positivity-preserving scheme which is also available to develop the theoretical analysis.The scheme has only cell-centered unknowns and preserves the local flux continuous,which is suitable for arbitrary star-shaped meshes.Moreover,the existence of the discrete solution for the nonlinear scheme is proved by using Brouwer's fixed-point theorem.Numerical results are presented to show that the scheme obtains second-order accuracy.In the second part,first we discuss the discrete scheme for diffusion problem with time-derivative term of conservative nonlinear energy.By developing new reasoning techniques,we overcome the difficulty caused by the time-varying nonlinear energy term,and prove rigorously the existence,uniqueness,convergence and stability of the solution of the fully implicit finite d-ifference scheme.Then we discuss the nonlinear iterative methods for solving the fully implicit(FI)scheme of equilibrium radiation diffusion problem.Together with the Picard factoriza-tion(PF)iteration method,three new nonlinear iterative methods,namely,the Picard-Newton factorization(PNF),Picard-Newton(PN)and derivative free Picard-Newton factorization(DF-PNF)iteration methods are studied,in which the resulting linear equations can preserve the parabolic feature of the original PDE.By using the induction reasoning technique to deal with the strong nonlinearity of the problem,rigorous theoretical analysis is performed on the fun-damental properties of the four iteration methods.It shows that they all have first-order time and second-order space accuracy,and moreover,can preserve the positivity of solutions.It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively.Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.In the third part,first we discuss the nonlinear fully implicit finite volume scheme for nonlinear diffusion problem.By giving a thorough estimate on the nonlinear discrete diffusion operator with weighted harmonic average diffusion coefficient,we analyze the compatibility of the discrete scheme.Moreover,the existence of the solution for the nonlinear scheme is proved by using Brouwer's fixed-point theorem.The convergence of the scheme is proved by using the estimates acquired by the existence property and a series of new reasoning techniques.Then we apply the fully implicit finite volume scheme to solve the equilibrium radiation diffusion equation based on a Saha ionization model.Considering the characters of the problem,in the designing of the iteration schemes,main attentions are focused on the discretization of the time derivative term.The time derivative term is divided into two parts and three kinds of iterations are provided accordingly,namely:Picard factorization iteration+Picard iteration(PF+Picard),Picard-Newton factorization iteration+Picard-Newton iteration(PNF+PN),PN iteration+PN iteration(PN+PN);Picard iteration is applied to the spatial derivative term.Numerical experi-ments show that the three iterative schemes all have second order space accuracy.