Nonlinear Spectral Analysis in a Multiply-Connected Region

谱差分格式是求解非线性偏微分方程，如Navier-Stokes方程的一种非常有效的方法。本文利用双周期函数的保角变换，将问题的有效性加以扩展，目标方程主要是带边值和/或初值的偏微分方程，本文处理的是三维连通区域或轴对称双连通区域。在第一阶段，假设流体是牛顿流体。(i)介绍了各种保角映射函数。(ii)处理各种配置。(iii)讨论了无限延伸、非等温边界和自然对流区域，结果表明，在谱差分格式下，利用Jacobian椭圆函数的保角变换在多连通区域是非常有效的。

The spectral difference scheme is very effective to numerical analysis for nonlinear partial differential equations, such as Navier-Stokes equations. Effectiveness has been extended, using a conformal mapping with doubly-periodic functions.Target equations are mainly partial differential equations with boundary values and/or initial values.In this project, triply-connected two-dimensional regions or axisymmetric doubly-connected regions are treated. At the first stage, fluid is assumed to be Newtonian.(i) Various conformal mapping function are introduced.(ii) Various configurations are treated.(iii) Region of infinite extension, non-isothermal boundary, and natural convection is treated.As a result, conformal mapping using Jacobian elliptic functions is found to be very effective in multiply-connected region under a spectral finite difference scheme.