Coordinate Transformations

Abstract

Partial differential equations in the physical domain X ⁿ can be solved on a structured numerical grid obtained by mapping a reference grid in the logical region Ξ ⁿ into X ⁿ with a coordinate transformation x(ξ) : Ξ ⁿ → X ⁿ. The structured grid concept also gives an alternative way to obtain a numerical solution to a partial differential equation, by solving the transformed equation with respect to the new independent variables ξ ⁱ on the reference grid in the logical domain Ξ ⁿ. Some notions and relations concerning the coordinate transformations yielding structured grids are discussed in this chapter. These notions and relations are used to represent some conservation-law equations in the new logical coordinates in a convenient form. The relations presented will be used in Chap. 3 to formulate various grid properties.