Nonlinear radiative diffusion equation is coupled with fluid equations or magnetic hydrodynamical equations in radiation hydrodynamics simulations. The time scales of radiation and fluid can be different. To use the same time step in the coupled system, one needs to design numerical schemes for the nonlinear radiative diffusion equation that are stable for large time steps. Due to the nonlinearity and degeneracy, fully implicit schemes may exhibit time step constraints. We propose and test two different semi-implicit schemes in this paper. One allows for a hyperbolic CFL condition, and only one linear diffusion system and a scalar forth order polynomial on each space grid have to be solved implicitly in each time step. The other has a much larger CFL number, but a nonlinear system has to be solved in each time step. Though the computational cost of the second scheme in each time step is higher, the overall computational cost can be lower. Extensive numerical examples for both smooth and compactly supported solutions are presented to show the accuracy and stability of the schemes.

在辐射流体动力学模拟中，非线性辐射扩散方程与流体方程或电磁流体动力学方程耦合。辐射和流体的时间尺度可以不同。为了在耦合系统中使用相同的时间步长，需要为非线性辐射扩散方程设计数值方案，该方案对于较大的时间步长是稳定的。由于非线性和简并性，完全隐式的方案可能会表现出时间步长约束。我们提出并测试了两种不同的半隐式方案。一个允许双曲CFL条件，并且在每个时间步中仅隐式求解每个空间网格上的一个线性扩散系统和一个标量四阶多项式。另一个具有更大的CFL数，但是必须在每个时间步中求解非线性系统。尽管第二方案在每个时间步中的计算成本较高，但总体计算成本却可以较低。给出了光滑和紧凑支持的解决方案的大量数值示例，以显示方案的准确性和稳定性。