Polynomial interpolation schemes for internal derivative distributions on structured grids

Abstract

Computational fluid dynamics problems which use a streamfunction formulation require differentiation to obtain the velocity field. If the velocity components are required at internal points on the grid (for example, for particle trajectory calculations) this necessitates some form of interpolation. Similarly, N-body computations which include self-gravity and cannot afford the luxury of an O( N 2) 1 r 2 calculation, require interpolation to obtain the force components at locations internal to an overlaid grid. These two independent examples reduce to the same generalized problem. Given a function distribution on a structured grid, which interpolation schemes give the most suitable solutions in terms of accuracy, computational efficiency and smoothness properties? The best numerical schemes are those which use higher order finite difference approximations for the derivatives (but not vast templates), perform interpolation directly over the derivative (and not the function), and have an odd power as the leading term in the polynomial expansion for the derivative. These methods include forms of bicubic and quintic-cubic interpolation but although generating accurate and realistic internal derivative estimates, come at a higher computational price. For simulations where the interpolation routines need to be called frequently, the lower order schemes such as cubic-linear and biquadratic interpolation, although less accurate, are more computationally efficient and may be more appropriate.