Backward Error Analysis for Numerical Integrators Backward Error Analysis for Numerical Integrators

We consider backward error analysis of numerical approximations to ordinary diie-rential equations, i.e., the numerical solution is formally interpreted as the exact solution of a modiied diierential equation. A simple recursive deenition of the modiied equation is stated. This recursion is used to give a new proof of the exponentially closeness of the numerical solutions and the solutions to an appropriate truncation of the modiied equation. We also discuss qualitative properties of the modiied equation and apply these results to the symplectic variable step-size integration of Hamiltonian systems, the conservation of adiabatic invariants, and numerical chaos associated to homoclinic orbits.