V. Truncation Error

Difference representations on curvilinear coordinate systems are constructed by first transforming derivatives with respect to cartesian coordinates into expressions involving derivatives with respect to the curvilinear coordinates (the metric coefficients). The derivatives with respect to the curvilinear coordinates are then replaced with difference expressions on the uniform grid in the transformed region. The "order" of a difference representation refers to the exponential rate of decrease of the truncation error with the point spacing. On a uniform grid this concerns simply the behavior of the error as the point spacing decreases. With a nonuniform point distribution, there is some ambiguity in the interpretation of order, in that the spacing may be decreased locally either by increasing the number of points in the field or by changing the distribution of a fixed number of points. Both of these could, of course, be done simultaneously, or the points could even be moved randomly, but to be meaningful the order of a difference representation must relate to the error behavior as the point spacing is decreased according to some pattern. This is a moot point with uniform spacing, but two senses of order on a nonuniform grid emerge: the behavior of the error (1) as the number of points in the field is increased while maintaining the same relative point distribution over the field, and (2) as the relative point distribution is changed so as to reduce the spacing locally with a fixed number of points in the field.