Numerical Astrophysics Using Supercomputers and Adaptive Scheme

我们能够在天体物理学领域使用几种超级计算机。它们是向量并行机、大规模并行机和专用机。为了充分发挥它们的作用，我们必须选择最有效的数值方法。虽然数值天体物理学的研究领域很广，从太阳系到宇宙，都有一个共同的瓶颈，要想取得突破，就必须突破这个瓶颈。磁流体动力学方程是天体物理学中最重要的方程之一。非定常流体动力学方程是双曲型的。在这种情况下，它表明，数据很容易划分和分配到一些处理器。另一方面，椭圆型方程，如泊松方程，也是天体物理中最重要的方程之一，其并行化程度很低。这主要是由于系数矩阵的预处理很难并行化。3.作为一个泊松求解器，我们发现“多重网格迭代法”是快速相比，普通的求解器，这是很差的并行化。我们已经测量了用这种方法求解一次泊松方程所需的CPU时间。计算结果表明，嵌套网格技术对于求解大空间动态范围的问题是非常有效的。它使用了大量的网格;粗网格覆盖了整个数值框，而细网格覆盖了需要精细空间分辨率的区域。5.为了从嵌套网格方案到完全自适应的方案，我们必须开发一种创建/销毁网格的方法。这就要求代码理解哪个部分应该被仔细计算。

We are able to use several kinds of supercomputers in the field of astrophysics. They are vector-parallel machines, massive parallel machines, and special purpose machines. To bring their full play, we have to choose the most effective numerical method. Although the field of numerical astrophysics is wide : from the solar system to the universe, there must be a common bottleneck which should be broken to make a breakthrough. This project were planned to answer the above point.1.The (magneto-) hydrodynamics equation is one of the most important ones in the astrophysics. Equations for the unsteady hydrodynamics are hyperbolic-type. In this case, it is shown that data are easily divided and allocated onto a number of processors. We are able to parallelize the code and achive high performance.2.On the other hand, the elliptical-type equation, such as the Poisson equation, which is also one of the most important ones in the astrophysics, is poorly parallelized. This is mainly due to the fact that it is difficult to parallelize the preconditioning of the coefficient matrix.3.As a Poisson solver, we have found that "multigrid iteration method" is fast compared to the ordinary solvers which were poorly parallelized. We have measured the CPU time necessary to solve the Poisson equation once by this method. It is only a half of the time required to solve the magnetohydrodynamics equations for one time step.4.It is shown that the nested grid technique is very efficient for a problem with a large spatial dynamic range. It uses a number of grids ; coarser grids covers a whole numerical box, while finer ones cover a region which require a fine spatial resolution.5.To proceed from the nested grid scheme to a fully adaptive one, we have to develop a method of creation/destruction of the grids. This requires the code to understand which part should be calculated closely.