The Positivity- And Extremum-preserving Finite Volume Schemes For Radiation Diffusion Equations

The numerical simulation of radiation diffusion equations has great significance to the study of practical engineering and physical problems,such as confinement fusion,plasma physics,astrophysics and so on.Since the high resolution of interface is required in engineering,the Lagrangian scheme is usually used to simulate the compressible radiation hydrodynamics problems.The mesh of numerical simulation may be distorted with the movement of fluid,so the numerical simulation of energy diffusion needs to be carried out on distorted mesh.In this thesis,the positivityand extremum-preserving finite volume schemes for radiation diffusion equations on distorted meshes will be considered,and the main research work is as follows:First of all,we construct the positivity-preserving finite volume scheme for three-dimensional diffusion equations.The auxiliary unknowns are defined at cell vertexes and the primary unknowns are defined at cell centers.The discretization of diffusion flux is based on the classical nonlinear two-point flux approximation.A new positivity-preserving interpolation algorithm is constructed to avoid the negative value of auxiliary unknown.This algorithm is applicable to arbitrary polyhedral mesh and has second-order accuracy.In order to avoid the search process of convex decomposition of co-normal vector,we present a fixed stencil positivity-preserving finite volume scheme.The discrete expression of diffusion flux on each cell facet contains all auxiliary unknowns on the facet.We design a new interpolation algorithm and improve the Anderson acceleration algorithm to raise the computational efficiency of nonlinear iteration.Numerical examples illustrate that the new schemes in solving the discontinuous anisotropic diffusion problems on distorted meshes have second-order accuracy.To construct an extremum-preserving finite volume scheme for the unsteady diffusion equations,we introduce the harmonic average points to define the auxiliary unknowns,which simplifies the interpolation procedure.Since there is only one harmonic average point on each cell edge,the discrete expression of diffusion flux is defined on a small stencil.We prove that the solution of discrete scheme satisfies the extremum principle,and present stability analysis of the scheme.Secondly,the positivity-and extremum-preserving finite volume schemes are constructed to solve the two-and three-dimensional convection-diffusion equations.The positivity-preserving scheme is constructed based on the classical nonlinear twopoint flux approximation and the improved discretization method of convection flux.According to the linear reconstruction method which is used in the discretization of convection flux,a new interpolation algorithm is obtained.Since the convergence of Picard iteration method is slow in solving the nonlinear system,the improved Anderson acceleration algorithm is adopted to accelerate the convergence of nonlinear iteration.A fixed stencil positivity-preserving finite volume scheme is established for the radioactive waste repository problem in practical application.To assure the positivity-preserving property of the discrete scheme,a positive term is added in the discretization process of convection flux.This scheme does not need the positivitypreserving interpolation algorithm and the convex decomposition of co-normal vector.These properties make our scheme can effectively solve the radioactive waste repository problem on unstructured meshes.Besides,we present the second-order accurate extremum-preserving finite volume scheme for two-and three-dimensional convection-diffusion equations and prove that the solution of discrete scheme satisfies the extremum principle.The numerical experiments also verify the effectiveness of the scheme.Then,the positivity-preserving finite volume scheme of the non-equilibrium radiation diffusion equations is constructed.The cell vertexes are used to define the auxiliary unknowns,and the cell centers are applied to define the primary unknowns.The discrete expression of diffusion flux is defined on a fixed stencil.With the fluxlimited non-equilibrium radiation diffusion problem,the least square method is used to get the gradient of the numerical solution on distorted meshes,and then the high precision approximation of radiation diffusion coefficient is obtained.These properties enable our numerical scheme to effectively solve the non-equilibrium radiation diffusion equations on distorted meshes.The positivity-preserving and existence of solution for discrete scheme are theoretically proved.Finally,the positivity-and extremum-preserving finite volume schemes are presented for three-temperature radiation diffusion equations.With the severe discontinuity,strong nonlinearity and tightly coupled of the equations,it is a challenge to construct the numerical scheme for solving the three-temperature radiation diffusion equations on distorted meshes.By using the new nonlinear two-point flux approximation to discretize the diffusion flux,this scheme does not require the interpolation method to be positivity-preserving and the convex decomposition of normal vector on each cell edge.The existence of discrete solution for the positivity-preserving scheme is proved,and the stability analysis is also presented.The extremumpreserving finite volume scheme of three-temperature radiation diffusion equations is constructed based on the positivity-preserving scheme.This scheme satisfies the local conservation and extremum principle on distorted meshes.In addition,the existence of discrete solution and stability for extremum-preserving scheme are analyzed.The numerical results illustrate that the positivity-and extremum-preserving finite volume schemes can effectively solve the three-temperature radiation diffusion equations.