Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation

This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) 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 fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, 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.