The 2D 3-T radiation diffusion equations are typical strongly nonlinear partial differential equations widely applied in inertial confinement fusion, astrophysical radiative shock simulations and so on. Conventional numerical methods for such radiation equations take radiation temperature as the unknown variable, which suffer from poor iterative convergence and low numerical precision under strong nonlinear conditions. To overcome this difficulty, we propose a novel framework based on the radial basis function meshfree method, which takes radiative energy density as the primary unknown variables. Three time discretization schemes, namely the fully implicit scheme, backward differentiation formula scheme, and Crank-Nicolson scheme, are comprehensively evaluated, and the simplified Newton iteration is adopted for nonlinear solving. Numerical examples verify that the framework based on radiative energy density yields better nonlinear convergence and numerical precision compared with the conventional formulation based on radiation temperature. We compared the numerical results with those obtained by the finite element method, and good numerical agreement was achieved. We further investigated the energy conservation property of the proposed method. The numerical results showed that our method facilitates long-time simulations and exhibits favorable computational performance for multi-material problems.
2次元3温度辐射扩散方程是典型的强非线性偏微分方程，广泛应用于惯性约束聚变、天体物理辐射激波模拟等领域。针对此类辐射方程的传统数值方法以辐射温度作为未知变量，在强非线性条件下存在迭代收敛性差、数值精度低的问题。为克服这一困难，我们提出了一种基于径向基函数无网格方法的新型框架，以辐射能量密度作为主要未知变量。本文综合评估了三种时间离散格式，即全隐式格式、后向微分公式格式和Crank-Nicolson格式，并采用简化牛顿迭代进行非线性求解。数值算例验证表明，与基于辐射温度的传统格式相比，基于辐射能量密度的框架具有更好的非线性收敛性和数值精度。我们将数值结果与有限元方法获得的结果进行了比较，两者取得了良好的一致性。我们进一步研究了所提方法的能量守恒性质。数值结果表明，我们的方法有利于长时间模拟，并在多材料问题中表现出良好的计算性能。