Generalization of a Spectral Finite Difference Scheme

在目前的谱有限差分格式中，假设因变量表示为完全谱展开(在一个空间分量中)，如傅立叶展开，这导致在垂直于展开方向(S)的空间和时间上建立一个联立偏微分方程组。结果表明，该格式对原偏微分方程组在空间分量上的分解没有引入误差，至少对简单解析函数表示的单连通区或双连通区的自然对流/强迫对流/非牛顿流体流动具有较高的空间分辨率和计算速度。在目前的项目中，建议引入统一泛函，在二维单连通区域的情况下，该泛函将边界映射到圆。对于矩形等边三角形腔、矩形腔和多边形腔等几种复杂构型，得到了具体的映射解析函数。对于几种情况，映射用雅可比椭圆函数表示。结果表明，对于不同的参数组合，如Grashof数、Prandtl数和弹性数(对于粘弹性流体)，目前的方法至少在层流自然对流条件下是有效的。

In the current spectral finite difference scheme, dependent variables are assumed to be expressed in a complete spectral expansion (in one spatial component) such as Fourier expansion, which leads to a system of simultaneous partial differential equations in space normal to the direction (s) of expansion and in time. As a result, no error is introduced in decomposing the original partial differential equations in spatial components, so that the scheme possesses better resolution in space and high computation speed in nature at least for natural convection/forced convection/non-Newtonian fluid flow in a simply-connected region or over a doubly-connected region expressed in terms of a simple analytic function. Under the current project, proposed is introduction of a unified functional which maps the boundary to a circle in case of a two-dimensional simply-connected region. For several complex configuration, e.g. a. rectangular equilateral triangle cavity, a rectangular cavity, and a polygon cavity, a concrete mapping analytic function is obtained. For several cases mapping is expressed in terms of Jacobian elliptic functions. As a result, current schemes are found to be effective to get a steady-state solution at least under laminar natural convection for various combinations of parameters such as a Grashof number, a Prandtl number, and an elastic number (for a viscoelastic fluid).