辐射流体力学保物理特性的并行格式研究

The diffusion equation of radiation hydrodynamic is usually discreted by implicit scheme, which leads that the cost of computation is huge. Hence, we must use massive parallel computers to solve this problem. However, the implicit scheme leads to strong coupling of data, and the algorithm lacks the intrinsic parallelism, which becomes the bottleneck problem of parallel computation of radiation hydrodynamic numerical simulation. The parallel scheme can decouple these global data, and the parallel scalability is very high. Hence, it is necessary to construct the parallel scheme with intrinsic parallelism. Many literatures show that a scheme preserving some important properties of partial differential equation is usually better than a scheme without preserving these properties, which include conversation, positivity and maximum principle etc. In order to implement high efficient numerical simulation of radiation hydrodynamic problem, it is necessary to construct parallel scheme preserving some important properties. We aim to research the conservative parallel scheme of radiation hydrodynamic problem, and we expect that these schemes have high accuracy, and are high efficient and unconditional stable, moreover, they can preserve the positivity and maximum principle. Then we will give some theoretical results. We will research the cell-centered scheme, and consider how to give the expression of flux and the method of prediction and correction on the interior interface. At the same time, we will implement the parallel scheme in application program, and improve the computational efficient of numerical simulation.

辐射流体力学方程组中的扩散方程一般采用隐式格式离散，其计算量极大，必须采用大规模并行计算。但隐式格式导致数据强相关，算法缺乏内在并行度，这成为制约实现高效并行计算的瓶颈问题。并行格式能消除数据的整体相关性，并行可扩展性非常好。因此，需要研究具有并行本性的并行计算格式。已有的研究表明，保持偏微分方程性质的离散格式一般要比不能保持或保持较少的离散格式要好，这些性质包括守恒性、保正性、极值原理等。为实现辐射流体力学问题的高效数值模拟，需要研究保物理特性的并行格式。本项目将针对辐射流体力学问题，研究能够适应大规模并行机的高效、高精度、无条件稳定的守恒型并行计算格式，且格式能保持解的非负性，或满足极值原理，并给出相关的理论结果。在格式设计中，本项目将考虑单元中心型格式，重点研究如何合理地给出法向通量的离散表达式和分区界面处的预估校正方法。同时将研制相关的程序模块，提高实际问题的计算效率。

辐射流体力学方程组中的扩散方程一般采用隐式格式离散，算法缺乏内在并行度，并行格式能消除数据的整体相关性，并行可扩展性非常好。因此，需要研究具有并行本性的并行计算格式。已有的研究表明，保持偏微分方程性质的离散格式一般要比不能保持或保持较少的离散格式要好，这些性质包括守恒性、保正性、极值原理等。为实现辐射流体力学问题的高效数值模拟，需要研究保物理特性的并行格式。.本项目首先研究了辐射扩散方程和对流扩散方程的保物理特性格式，包括保正格式和保极值原理格式，改进了已有保正格式和保极值格式的精度和健壮性。然后，研究了能够适应大规模并行机的高效、高精度、无条件稳定的守恒型并行计算格式，重点研究了如何合理地给出法向通量的离散表达式和分区界面处的预估校正方法，设计出了扭曲网格上辐射扩散方程的具有并行本性的保物理特性格式等。.总之，我们圆满完成了本项目的研究目标，改进了已有的保正格式和保极值原理格式，设计出了具有并行本性的保物理特性的并行格式，并发表多篇高水平论文。