Multi-off-grid methods in multi-step integration of ordinary differential equations

Abstract

New off-grid methods, that are not limited by the Dahlquist stability theorem, are introduced for the numerical integration of first- and second-order systems of differential equations. The methods are characterized by having all derivative evaluations performed at locations off the grid of final solution values. All on-grid solution values and off-grid derivative evaluations are used over m back steps to calculate the solution value at the next on-grid location. There are n off-grid derivative evaluations associated with each of these intervals located at fractional positions γ₁h, ..., γ_nh relative to the on-grid locations. Off-grid geometric parameters are found which give highly stable integrators of maximum possible order, O(h^m+mn+n-1). These integrators also have the property that their coefficients decrease geometrically with remoteness from the current interval of integration; and although the second mean value theorem is not applicable in obtaining an exact error expression for these methods, the error coefficients are generally several orders of magnitude less than comparable on-grid Adams or Cowell error coefficients. Comparisons are made with other integration techniques for satellite orbital calculations and other systems or ordinary differential equations.

Work supported by NASA Goddard Space Flight Center Under Contract No. NAS 5-11790