Spatial interpolation methods for integrating Newton's equation

Numerical integration of Newton's equation in multiple dimensions plays an important role in many fields such as biochemistry and astrophysics. Currently, some of the most important practical questions in these areas cannot be addressed because the large dimensionality of the variable space and complexity of the required force evaluations precludes integration over sufficiently large time intervals. Improving the efficiency of algorithms for this purpose is therefore of great importance. Standard numerical integration schemes (e.g., leap-frog and Runge-Kutta) ignore the special structure of Newton's equation that, for conservative systems, constrains the force to be the gradient of a scalar potential. We propose a new class of "spatial interpolation" (SI) integrators that exploit this property by interpolating the force in space rather than (as with standard methods) in time. Since the force is usually a smoother function of space than of time, this can improve algorithmic efficiency and accuracy. In particular, an SI integrator solves the one-and two-dimensional harmonic oscillators exactly with one force evaluation per step. A simple type of time-reversible SI algorithm is described and tested. Significantly improved performance is achieved on one-and multi-dimensional benchmark problems.