ERROR ANALYSIS AND DIFFERENCE EQUATIONS ON CURVILINEAR COORDINATE SYSTEMS

Abstract

This chapter discusses methods for deriving difference equations on curvilinear coordinate systems and the effect of coordinate systems on the solution. A computational grid must be constructed when solving partial differential equations by finite-difference or finite element methods. At present, there are many grid generation algorithms. The choice of algorithm will depend on the users' desire for control over properties, such as orthogonality of coordinate lines, location of grid points, and smoothness of grid point distribution. All of these properties may affect the accuracy of the numerical solution. Thereafter, the interpolation technique may affect the local truncation error as well as the convergence rate of iterative algorithms and the stability of explicit algorithms. A curvilinear coordinate system in the xy-plane is understood to be the image of a rectangular Cartesian coordinate system in a ξη-plane. The induced grid is, therefore, composed of quadrilateral cells and difference equations may be derived by transforming the partial differential equation to the ξη-plane. The degree of skewness in a nonorthogonal coordinate system must be limited to maintain the order of the numerical algorithm. An alternate method, the finite volume method, for approximating conservation laws has been widely used on curvilinear coordinate systems. A loss of accuracy in the numerical algorithm may occur if the grid is severely distorted. Therefore, when the region is too complicated, it may be advisable to partition the region and construct a separate curvilinear coordinate system for each subregion.