We present a JASMIN-based two-dimensional parallel implementation of an adaptive combined preconditioner for the solution of linear problems arising in the finite volume discretisation of one-group and multi-group radiation diffusion equations. We first propose the attribute of patch-correlation for cells of a two-dimensional monolayer piecewise rectangular structured grid without any suspensions based on the patch hierarchy of JASMIN, classify and reorder these cells via their attributes, and derive the conversion of cell-permutations. Using two cell-permutations, we then construct some parallel incomplete LU factorisation and substitution algorithms, to provide our parallel -GMRES solver with the help of the default BoomerAMG in the HYPRE library. Numerical results demonstrate that our proposed parallel incomplete LU preconditioner (ILU) is of higher efficiency than the counterpart in the Euclid library, and that the proposed parallel -GMRES solver is more robust and more efficient than the default BoomerAMG-GMRES solver.

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations (Bco and Bco), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner Bcoα involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that Bcoα-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

We develop a Lions domain decomposition algorithm based on a cell functional minimization scheme on non-matching multi-block grids for nonlinear radiation diffusion equations, which are described by the coupled radiation diffusion equations of electron, ion and photon temperatures. The L2 orthogonal projection is applied in the Robin transmission condition of non-matching surfaces. Numerical results show that the algorithm keeps the optimal accuracy on the whole computational domain, is robust enough on distorted meshes and curved surfaces, and the convergence rate does not depend on Robin coefficients. It is a practical and attractive algorithm in applying to the two-dimensional three-temperature energy equations of Z-pinch implosion simulation.