Construction of Numerical Analysis for High-performance Large-Scale Computation

1.在设计流动问题的数值格式时，如何逼近对流撕裂是一个关键问题。特征有限元近似基于材料导数的近似，导数是时间导数项和对流项的和。到目前为止，已经采用了时间增量一阶精度的特征线法有限元格式。我们发展了一个在时间增量上具有二阶精度的有限元格式，并得到了可能的最佳误差估计。该格式在数值积分误差方面比一阶格式更稳健，能够更稳定、更准确地求解流动问题。2.我们发展了一种有限元格式，并建立了具有温度相关粘性的对流问题的误差估计。地球上的地幔对流和熔化的玻璃对流等热传导问题的粘度强烈依赖于温度。…此外，这种相关性在对流图案的形成中起着重要的作用。我们的格式适用于具有温度相关的粘度、导热系数和热膨胀系数的一般Rayleigh-Benard问题。利用该格式对三维球壳中的地幔对流进行了大规模的数值模拟，成功地得到了复杂的对流换热图样。3.在无限精度计算中，我们在一台具有10CPU和20 GB内存的高性能计算机集群上成功地进行了大规模并行计算。对于一维边值问题，得到了精度为4995位的非常精确的结果。我们使用该系统对反问题进行了直接的数值模拟，使反问题的数值分析成为可能。4.建立了磁矢势和电标势中的涡流问题的数学模型，并采用分层区域分解方法进行了求解。该解决方案在并行计算环境下是有效的。利用这种方法，我们对磁矢势中的非线性静磁问题进行了大规模的数值模拟。较少

1.In devising numerical schemes for flow problems, how to approximate the convection torn is a crucial point. Characteristic finite element approximation is based on the approximation of the material derivative, which is the sum of the time derivative term and the convection term. So far finite element schemes of characteristic method of the first-order accuracy in time increment have been used. We have developed a finite element scheme of the second-order accuracy in time increment and obtained the best possible error estimate. This scheme is more robust than the first-order scheme with respect to numerical integration error and can solve flow problems more stably and accurately.2.We have developed a finite element scheme and established an error estimate for heat convection problems with temperature-dependent viscosity. The viscosity of heat conduction problems such as mantle convection in the Earth and melting glass convection in the furnace is strongly dependent on the temperature. … More The dependence plays an important role in die formation of convection patterns. Our scheme is applicable for the general Rayleigh-Benard problems with temperature-dependent viscosity, thermal conductivity, and thermal expansion coefficient. Using this scheme we have carried out large-scale numerical simulation of Earth's mantle convection in three-dimensional spherical shell and succeeded in obtaining complex heat convection patterns.3.In the infinite precision computation we have succeeded a large-scale parallel computation using a cluster of high-performance computers with 10CPU and 20GB memory. For one-dimensional boundary-value problems very precise results with precision 4995 digits have been obtained. We have used this system to perform direct numerical simulation of inverse problems and made possible a numerical analysis of inverse problems.4.Formulating eddy current problems in magnetic vector potential and electric scalar potential, we have solved them using a hierarchical domain decomposition method. This solution has been shown to be effective under the environment of parallel computation. By this method we have carried out large-scale numerical simulation of nonlinear static magnetic problems in magnetic vector potential. Less