Error Induced by Coordinate Systems,
MISSISSIPPI STATE UNIV MISSISSIPPI STATE DEPT OF MATHEMATICS
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The choice of a curvilinear coordinate system can have a substantial effect on the error in the numerical solution of a partial differential equation. The truncation error is dependent not only on the higher order derivatives of the solution and the local grid spacing, but also on the rate-of-change of the grid spacing and on the departure of the grid from orthogonality. The coordinate system influences the smoothness as well as the accuracy of the numerical solution. This fact is evident by recalling the general principal that the smoothness of the solution of a partial differential equation depends on the smoothness of the coefficients. The coefficients of the equations, in terms of curvilinear coordinates, depend on the derivatives of the functions defining the coordinate system. Examples where lack of smoothness can be traced to the coordinate system appear in papers. This report will analyze the local truncation error in the approximation of first and second order derivatives on a curvilinear grid. Standard second order central differences have been used. An analogous development could be carried out using one-sided or higher order difference approximations. While only two-dimensional grids are considered most results can be extended in an obvious manner to three dimensions.