Compact, implicit difference schemes for a differential equation’s side conditions

Abstract:

Lynch and Rice have recently derived compact, implicit (averaged-operator) difference schemes for the approximate solution of an mth order linear ordinary differential equation under m separated side conditions. We construct here a simpler form for a compact, implicit difference scheme which approximates a more general side condition. We relax the order of polynomial exactness required for such approximate side conditions. We prove appropriate convergence rates of the approximate solution (and its first $m - 1$ divided differences) to (those of) the solution, even, of multi-interval differential equations. Appropriate, here, means kth order convergence for schemes whose interior equations are exact for polynomials of order $k + m$ and whose approximation of a side condition of order l is exact for polynomials of order $k + l$. We also prove the feasibility of shooting (and of multiple shooting) based on initial divided differences. The simplicity of the proofs is based upon the simplicity of form of the approximating side conditions, together with the crucial stability result of Lynch and Rice for their interior difference equations under divided-difference initial data.