  The radiation diffusion problem is a kind of {time-dependent} nonlinear
equations. For solving the radiation diffusion equations, many linear systems
are obtained in the nonlinear iterations at each time step. The cost of linear
equations dominates the numerical simulation of radiation diffusion
applications, such as inertial confinement fusion, etc. {Usually,} iterative
methods are used to solve the linear systems in {a} real application.
{Moreover, the solution of the previous nonlinear iteration or the solution of
the previous time step is typically used as the initial guess for solving the
current linear equations.} Because of the strong local character in ICF, with
the advancing of nonlinear iteration and time step, the solution of the linear
system changes dramatically in some local domain, and changes mildly or even
has no change in the rest domain. In this paper, a local {character-based}
method is proposed to solve the linear systems of radiation diffusion problems.
The proposed method consists of three steps: firstly, a local domain (algebraic
domain) is constructed; secondly, the subsystem on the local domain is solved;
and lastly, the whole system will be solved. Two methods are given to construct
the local domain. One is based on the spatial gradient, and the other is based
on the residual. Numerical tests for a two-dimensional heat conduction model
problem, and two real application models, the multi-group radiation diffusion
equations and the three temperature energy equations, are conducted. The test
results show that the solution time for solving the linear system can be
reduced dramatically by using the local character-based method.
