多介质非线性能量方程组是应用领域中描述能量传输过程的偏微分方程组。本项目将针对这一复杂的具有间断系数的非线性偏微分方程组，研究适用于求解一般物态方程的拉氏多介质辐射热传导问题的计算方法。将设计新的计算方法，包括：(i) 对于热传导项，设计适用于大变形网格的单元中心型守恒离散格式，使其能够抑制非物理数值振荡且具有高于一阶的精度； (ii) 对于能量的时间变化率，设计适用于实际的非线性能量函数的离散方法，使其能够提高能量方程求解的守恒性；(iii) 设计高效的非线性迭代方法，使其不仅减少守恒误差，而且能够提高迭代求解的健壮性、缩短实际求解时间。解决复杂网格上多介质非线性能量方程组现有的计算方法存在的两个关键的困难问题：一是出现非物理振荡和精度低的问题；二是求解效率低甚至失效和守恒性差的问题。并研制高效的解法器，应用于实际问题求解，验证计算方法的精度和健壮性。

The system of nonlinear multi-material energy equations is a system of partial differential equations, which describes energy transport process in applications. For solving numerically the complex system of nonlinear partial differential equations with discontinuous coefficients, this project will construct accurate and efficient computational methods for solving radiation conduction problems with general equation of state on Lagrangian meshes. New methods will be proposed, including (i) cell-centered conservative schemes on distorted meshes for heat conduction term, which can restrain numerical oscillations and get higher accuracy than first order; (ii) discrete methods of practical nonlinear energy functions for the time increment rate of energy, which can assure to get better conservation of energy; (iii) efficient methods of nonlinear iteration, which not only decrease conservative error, but also increase robustness of iteration method and decrease practical run time. Our new methods will settle two key and difficult issues on numerical methods for nonlinear multi-material energy equations on complicated meshes: (i) low accuracy and unphysical oscillation which are produced by existing discrete schemes; (ii) low efficiency even failure and large conservative error which are generated by exiting nonlinear iteration methods. Fast and accurate solvers will be developed and applied for numerically simulating practical radiation hydrodynamic problems, and the accuracy and robustness of the numerical methods will be demonstrated.