We developed a high performance algebraic solver for nonlinear systems discretized from two-dimensional energy equations with three temperatures by a nine point scheme.The main idea is to solve the system by an inexact Newton method and preconditioned Krylov subspace methods in the frame of PNK and JFNK methods.Numerical experiments show the efficiency of the algebraic solvers.It is shown that our PNK method is 6 times faster than the nonlinear block Gauss-Seidel method. The JFNK and PNK methods are also compared.

In a sparse linear system derived from two-dimensional three-temperature energy equations, the diagonal dominan varies greatly from row to row and so is the magnitude of the elements. We provide a new scaling method to improve the diagonal dominance. As ILUT is used to the derived linear system, it reserves the number of elements in each row and several relatively large elements related to the photon are dropped due to the large difference among elements. To improve the equality of the ILUT, we provide a new method named multiple row ILUT (MRILUT), in which multiple rows are computed before dropping. The provided methods are embedded into a preconditioned Krylov subspace method to solve the actual two-dimensional energy equations with three temperatures. The number of iteration at each time step and the total computation time are both greatly reduced.

A new type of hybrid iterative method is presented, which is competent in solving the large scale sparse linear systems derived from the 2-dimensional 3-temperature radiation dynamic energy equations. Numerical results show that the new method is as 4 times fast as the old ones.Especially on the cases the old one doesn't converge, the new method can easily get the solution to the precision required. It can successfully complete the simulation and the final physical parameters of the simulation coincide with the theory.

It presents an effective fill-in technology for the preconditioners of large sparse linear algebraic equations arising from the difference discretizations of high dimensional physical problems, and discusses the relations between fill-in and numerical costs. By using this technology into the practical problems for verification, numerical results obtained are well coincided with the theoretical analyses.

To solve complex systems obtained from 2D nonlinear Schrödinger equation, a method of high-order PCG coupling with BICG has been developed. Meanwhile, the complex systems of order M can be reconstructed as the real systems of order 2M. From the theory of order matrix, the high order approximate LU decompositions are given as the preconditioners to solve the real systems. Numerical results show that the computational efficiency can be nearly doubled by the high-order PCG method comparing with the ICCG method.