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

三个项目计划，结合数值解 分支理论偏微分方程 分析. 首先，Belousov-Zhabotinski的简化模型 反应，专为计算速度，是要研究。 第二，发展了一种新的数值方法， 大型刚性系统的集成。 该方法实现 不求矩阵求逆的稳定性 小Krylov矩阵 该方法将被修改以解决 Navier-Stokes方程及其性质将被分析 理论上 第三，现有的数字代码，解决了Boussinesq 圆柱体中的Rayleigh-Benard对流方程 几何，将用于研究打破轴对称， 快速旋转的影响，以及二维的有效性 用于图案形成的模型方程。 偏微分方程的数值解是 其本身和应用都很重要， 各种物理科学和工程应用。

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.