A numerical model of three-dimensional mantle convection ; the effects of strong temperature-dependence of viscosity

已经开发了一个数值代码的三维模拟地幔对流作为一个时间依赖的对流，具有强烈的温度依赖粘度的牛顿流体。在地幔对流模拟中，离散的动量方程和连续性方程形成了一套巨大的流体速度和压力线性方程组。由于系数矩阵依赖于时间，而粘度依赖于温度，对流依赖于时间，因此必须在每个时间步长完整而准确地求解庞大的线性方程。然而，由于系数矩阵的大小，在三维模拟中，线性方程的直接方法并不是一个实际的选择，必须采用迭代法。迭代法的问题是收敛速度慢。在三维模拟中，缓慢的收敛性实际上阻碍了基本方程的精确时间积分。在这里，我开发了一个有效的和精确的迭代求解器的动量和连续性方程设计的矢量并行计算机。我将动量和连续性方程的迭代求解器simple算法与泊松方程的直接求解器SEVP方法相结合，成功地求解了基本方程，其收敛速度比传统方法的收敛速度快30倍。我将数值代码应用于具有强烈温度依赖性粘度的流体的热对流的三维模拟，并成功地通过80,000个时间步准确地积分了基本方程。仿真中网格点个数为33x33x33，每个时间步cpu时间为5.4秒。

A numerical code has been developed for three-dimensional simulations of mantle convection as a time-dependent convection of a Newtonian fluid with a strongly temperature-dependent viscosity. In the simulations of mantle convection, the discretized momentum and continuity equations form a set of huge linear equations for fluid velocity and pressure. Since the coefficient matrix depends on time when the viscosity depends on temperature and the convection is time-dependent, the huge linear equation must be fully and accurately solved at each time-step. Because of the size of the coefficient matrix, however, direct methods of linear equations are not a practical choice in the three-dimensional simulation and iterative method must be employed. The problem in an iterative method has been its slow convergence. The slow convergence has practically inhibited accurate time-integration of the basic equations in three-dimensional simulations. Here, I developed an efficient and accurate iterative solver of the momentum and continuity equation designed for a vector parallel computer. I combined an iterative solver of momentum and continuity equations called SIMPLER algorithm with a direct solver of Poisson's equation called SEVP method and succeeded in solving the basic equations with a convergence rate 30 times higher than the convergence rate in the traditional methods. I applied the numerical code to a three-dimensional simulation of a thermal convection of a fluid with a strongly temperature-dependent viscosity and succeeded in accurately integrating the basic equations by 80,000 time steps. The number of mesh points was 33x33x33 and the cpu-time was 5.4 second for each time-step in the simulation.