Monotone Finite Volume Schemes And Picard-Newton Iteration Schemes For Radiation Diffusion Equations

In this thesis, monotone finite volume schemes and Picard-Newton nonlinear it-eration schemes are proposed for radiation diffusion equations. First, we propose a kind of positivity preserved decomposition of normal vector, and construct a nonlin-ear finite volume (FV) scheme for diffusion equation on star-shaped polygonal meshes which is proved to be monotone, i.e., it preserves positivity of analytical solutions for diffusion problems with strongly anisotropic and heterogeneous full tensor coefficients. The scheme has only cell-centered unknowns, treats material discontinuities rigorously, and offers an explicit expression for the normal flux. The coefficient matrix of linear system is sparse, and the number of nonzero entries is as small as possible. In the construction of discrete normal flux on each cell-edge, both the geometric character of distortion of cells and the feature of physical variables on that cell-edge are taken into account. Then, we construct a new monotone finite volume scheme for multimaterial non-equilibrium radiation diffusion problem on distorted quadrilateral meshes, and the scheme also takes account of the geometric distortion of cells and changes of physical variables on each edge, moreover, a part of the mesh stencil is fixed. Numerical re-sults demonstrate our monotone scheme doesn’t produce negative values on distorted meshes, and its accuracy is not lower than one order.Picard-Newton (P-N) and derivative free Picard-Newton (DFPN) nonlinear it-eration schemes are proposed for numerically solving the system of non-equilibrium diffusion equations, which are highly nonlinear and tightly coupled. Moreover, two time step control methods are investigated for P-N scheme and a study of temporal accuracy of a first order time integration is presented. The solution process of P-N scheme is as follows:firstly Newton linearization is applied for the flux-limited radia-tion diffusion model, which results in a system of convection-diffusion equations with tensor coefficients proved to be symmetric positive definite; then a cell-centered finite volume scheme is extended to discretize the radiation diffusion model with flux limiter on distorted quadrilateral meshes. Its main part is consistent with the well-known Pi-card iteration method, and P-N iteration method can be formulated by adding Newton correction terms to Picard iteration method. Our P-N method is different from previ-ous Newton methods by making different spatial discretization for Newton correction terms in linearized PDEs. The numerical results confirm the theoretical predications, and show that these schemes are robust and effective. Moreover, our P-N method needs not introduce more restricted time steps than that used for Picard method.