Computational and Bifurcation-Theoretic Study of Reaction- Diffusion and Navier-Stokes Equations

计划了三个方案，将偏微分方程组的数值解与分叉理论分析相结合。首先，要研究一个简化的Belousov-Zhabotinski反应模型，该模型是为计算速度而设计的。其次，发展了一种新的计算大型刚性系统时间积分的数值方法。这种方法通过对一个小的(Krylov)矩阵求幂来实现稳定性，而不需要求逆矩阵。将该方法改进为求解Navier-Stokes方程，并对其性质进行了理论分析。第三，现有的求解圆柱几何中Rayleigh-Benard对流的Boussinesq方程的数值程序，将被用来研究轴对称的破坏，快速旋转的影响，以及用于图案形成的二维模式方程的有效性。偏微分方程的数值解对于它们本身以及在各种物理科学和工程应用中的应用都是重要的。

Three projects are planned, combining the numerical solution of partial differential equations with bifurcation-theoretic analysis. First, a simplified model of the Belousov-Zhabotinski reaction, designed for computational speed, is to be studied. Second, a new numerical method has been developed for temporal integration of large stiff systems. This method achieves stability without matrix inversion by instead exponentiating a small (Krylov) matrix. The method will be modified to solve the Navier-Stokes equations and its properties will be analyzed theoretically. Third, an existing numerical code which solves the Boussinesq equations of Rayleigh-Benard convection in a cylindrical geometry, will be used to study the breaking of axisymmetry, the effects of rapid rotation, and the validity of two-dimensional model equations for pattern formation. Numerical solutions of partial differential equations are important both in their own right and in applications to a wide variety of physical science and engineering applications.