Convergence of LMM when the solution is not smooth

Abstract

LMM (linear multistep methods) are popular for the solution of the initial value problem for a system of ordinary differential equations. In the classical theory it is assumed that the solution is as smooth as necessary. LMM are constructed to be of about as high order as possible, subject to stability, so it is not obvious that they will provide reasonable results when the the solution is not as smooth as anticipated. It is shown that they do, the order is just reduced. The discretization error constants are investigated using Peano kernels. In practice, solutions seem to be piecewise smooth. It is shown then that the order of convergence is two higher than might be expected. Piecewise smooth solutions commonly arise when data is fitted with piecewise polynomial functions. An example of the propagation of sound in the ocean illustrates this and confirms the theory presented.