Generalization of a Spectral Finite Difference Scheme

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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).