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 preconditioner of incomplete factorization types and a modified version are provided for the 3-D problems with the help of local block decomposition of a block tridiagonal matrix recursively. Then the existence of both preconditioners is focused on. For the seven point matrix discreted from the 3-D Laplace operator with the seven point difference technique, the actual condition numbers are computed and the results show that for the non-modified preconditioner, the condition number is proportional to N^2／3, where N is the order of the matrix. For the modified version, the condition number is proportional to the cube root of the order of the matrix thereafter. Finally, highly efficient implementation is considered and several effective experiments are done for these preconditioners on personal computers with main frequency of 550MHz and memory of 256M.