Exponentially fitted methods that preserve conservation laws

The exponential fitting technique uses information on the expected behaviour of the solution of ordinary and partial differential equations to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In these cases, they perform better than standard methods, particularly in the large time-step regime. In this paper we consider exponentially fitted Runge-Kutta methods and we give characterizations of those that preserve local conservation laws of linear and quadratic quantities. As a benchmark for the general theory we apply the symplectic midpoint and its exponentially fitted version to approximate breather wave solutions of partial differential equations arising as models in several fields, such as fluid dynamics and quantum physics. Numerical tests show that the exponentially fitted method performs better and its solution exactly satisfy discrete conservation laws of mass (or charge) and momentum.