In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit methods require impractically small time steps, while implicit methods, which offer stability, come with the challenge to guarantee the convergence of nonlinear iterative solvers. To overcome these challenges, we propose a predictor-corrector approach and design proper implicit-explicit time discretizations. In the predictor step, the system is reformulated into a nonconservative form and linear diffusion terms are introduced as a penalization to mitigate strong nonlinearities. We then employ a Picard iteration to secure convergence in handling the nonlinear aspects. The corrector step guarantees the conservation of total energy, which is vital for accurately simulating the speeds of propagating sharp fronts in this system. For spatial approximations, we utilize local discontinuous Galerkin finite element methods, coupled with positive-preserving and TVB limiters. We validate the orders of accuracy, conservation properties, and suitability of using large time steps for our proposed methods, through numerical experiments conducted on one- and two-dimensional spatial problems. In both homogeneous and heterogeneous non-equilibrium radiation diffusion problems, we attain a time stability condition comparable to that of a fully implicit time discretization. Such an approach is also applicable to many other reaction-diffusion systems.

翻译：本文针对非平衡辐射扩散问题的数值模拟，发展了一类高阶守恒方法。该问题在数值求解中面临显著挑战，主要体现在刚源项中的强非线性以及非线性扩散项的退化性。显式方法需要极小的时间步长，而具有稳定性的隐式方法则面临非线性迭代求解器收敛性难以保证的难题。为克服这些挑战，本文提出一种预测-校正方法，并设计了合适的隐式-显式时间离散格式。在预测步骤中，将原方程重写为非守恒形式，并引入线性扩散项作为惩罚项以缓解强非线性，进而采用Picard迭代确保非线性求解的收敛性。校正步骤确保总能量守恒，这对精确模拟该系统中传播的尖锐前沿速度至关重要。在空间逼近方面，我们采用局部间断伽辽金有限元方法，并结合保正限制器和TVB限制器。通过一维和二维空间问题上的数值实验，验证了所提方法的精度阶、守恒特性以及采用大时间步长的适用性。在均匀介质和非均匀介质的非平衡辐射扩散问题中，该方法的时间稳定性条件与全隐式时间离散格式相当。本文方法同样适用于其他许多反应-扩散系统。