New Numerical Methods for Flow Problems and their Numerical Simulation

(1)发展了一种精确计算流体对浸没在流场中的物体的阻力和升力的数值方法。将这些值转化为区域上的等价积分，我们成功地建立了误差估计。用这种方法得到了球体的精确阻力系数。(2)具有无限Prandtl数的Rayleigh-Benard方程组是描述地幔对流等慢流中热对流现象的数学模型。我们为该系统开发了有限元格式，建立了误差估计，编制了有效的三维并行计算程序，并对地幔对流进行了数值模拟。(3)发展了数学理论和计算算法，从数值计算结果中求出偏微分方程组的精确解。我们将Dirichlet-Neumann映射应用于外问题，发展了求解调和方程的电荷模拟方法和Helmholtz方程的区域分解方法。(5)建立了近海石油污染对生态系统影响的数学模型，并进行了数值模拟和可视化。所得结果与物理实验结果吻合较好。(6)多孔介质流动的界面行为随初始状态的不同而不同。利用有限差分方法，我们给出了解的支撑点分离的充分条件和初始界面等待时间的上下限估计保持不变。

(1) We have developed a numerical method for computing accurately drag and lift exerted by the fluid to a body immersed in the flow field. Transforming those values to equivalent integrals in the domain, we have succeeded in establishing error estimates. We have obtained accurate drag coefficients of a sphere by this method. The restults have been extended to evolutional problems and best possible error estimates have been obtained.(2) The system of Rayleigh-Benard equations with infinite Prandtl number is a mathematical model describing heat convection phenomena in slow flows such as the Earth's mantle convection.We have developed a finite element scheme for this system, established error estimates, made an effective parallel code for three-dimensional computation, and performed numerical simulation for the Earth's mantle convection.(3) We have developed mathematical theory and computation algorithms to find exact solutions of partial differential equations from numerical computation results. Those methods have been applied to the stationary Stokes equations, the Navier-Stokes equations, stationary bifurcation solution of heat convection problems.(4) We have applied Dirichlet-Neumann mapping to exterior problems and developed combined numerical methods with a charge simulation method for the harmonic equation and with a domain decomposition method for the Helmholtz equation.(5) We have constructed a mathematical model to analyze effect to ecological system caused by coastal oil pollution and performed the numerical simulation and the visualization. The obtained results are in good agreement with physical experimental results.(6) The interface of porous media flow takes various behavior depending on the initial state. Using finite difference method we have given a sufficient condition for the separation of the support of the solution and upper and lower estimates of the waiting time for the initial interface to keep invariant.