  We present a new class of exponential integrators for ordinary differential
equations: locally exact modifications of known numerical schemes. Local
exactness means that they preserve the linearization of the original system at
every point. In particular, locally exact integrators preserve all fixed points
and are A-stable. We apply this approach to popular schemes including Euler
schemes, implicit midpoint rule and trapezoidal rule. We found locally exact
modifications of discrete gradient schemes (for symmetric discrete gradients
and coordinate increment discrete gradients) preserving their main geometric
property: exact conservation of the energy integral (for arbitrary
multidimensional Hamiltonian systems in canonical coordinates). Numerical
experiments for a 2-dimensional anharmonic oscillator show that locally exact
schemes have very good accuracy in the neighbourhood of stable equilibrium,
much higher than suggested by the order of new schemes (locally exact
modification sometimes increases the order but in many cases leaves it
unchanged).
