Monitoring energy drift with shadow Hamiltonians

The application of a symplectic integrator to a Hamiltonian system formally conserves the value of a modified, or shadow, Hamiltonian defined by some asymptotic expansion in powers of the step size. An earlier article describes how it is possible to construct highly accurate shadow Hamiltonian approximations using information readily available from the numerical integration. This article improves on this construction by giving formulas of order up to 24 (not just up to 8) and by greatly reducing both storage requirements and roundoff error. More significantly, these high order formulas yield remarkable results not evident for 8th order formulas, even for systems as complex as the molecular dynamics of water. These numerical experiments not only illuminate theoretical properties of shadow Hamiltonians but also give practical information about the accuracy of a simulation. By removing systematic energy fluctuations, they reveal the rate of energy drift for a given step size and uncover the ill effects of using switching functions that do not have enough smoothness.