Locally exact modifications of numerical integrators

We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing numerical schemes in order to make them locally exact. The resulting schemes preserve all fixed points and are A-stable. The most promising results concern the discrete gradient method (modified implicit midpoint rule) where we succeeded to preserve essential geometric properties and the final results have a relatively simple form. In the case of one-dimensional Hamiltonian systems numerical experiments show that our modifications can increase the accuracy by several orders of magnitude. The main result of this paper is the construction of energy-preserving locally exact discrete gradient schemes for arbitrary multidimensional Hamiltonian systems in canonical coordinates.