Error Growth in the Numerical Integration of Periodic Orbits, with Application to Hamiltonian and Reversible Systems

Abstract

We analyze in detail the growth with time (of the coefficients of the asymptotic expansion) of the error in the numerical integration with one-step methods of periodic solutions of systems of ordinary differential equations. Variable stepsizes are allowed. We successively consider "general," Hamiltonian, and reversible problems. For Hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated, numerical methods with relevant geometric properties (symplecticness, energy-conservation, reversibility) are proved to have better error growth than "general" methods.

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

“…As was shown in [5], this condition is satisfied when the function τ in (3.6) is such that τ (Y ) = τ (ΛY ). It is well known that symmetric FSLMM2s satisfy the condition that all the roots of the first characteristic polynomial R have modulus 1 [7].…”

“…From above we know that this term grows linearly, and therefore the result is proved for general y 0 . (For a more detailed discussion of the interpretation of the integrals above, see the remark after Theorem 5.1 in [5], as its proof is very similar to that one.) Remark 3.14.…”

“…We prove stability here for weakly as well as strongly stable methods assuming only some mild assumptions on the stepsizes and the variable coefficients. Secondly, asymptotic expansions of the global error on the stepsize or the tolerance have been proved in the literature for variable-stepsize one-step methods [5] and FSLMMs [6,11,14,18], but not for VSLMMs, which we do here. Both facts, the stability and the new asymptotic expansions, are stated in section 2 and then proved in the appendix for the sake of clarity.…”

“…On the other hand, as studied in [5], the Jordan structure of the monodromy matrix M 0 associated to (5.1) puts this problem under situation (R2 ). 5.1.…”

“…For variable-stepsize one-step methods, asymptotic expansions of the global error have been deduced in [5] which allow a study on the growth of error with time when integrating periodic orbits. In the particular case when the system is reversible, symmetric variablestepsize one-step methods have proven to be an excellent tool [1], [5], [13], [19]. However, the fact that the stepsize selection must also be symmetric makes these methods implicit, requiring special techniques to reduce the additional work.…”