Exponentially fitted methods that preserve conservation laws

TL;DR

This paper develops exponentially fitted Runge-Kutta methods that preserve local conservation laws in differential equations, especially for oscillatory solutions like wave equations, enhancing accuracy and physical fidelity.

Contribution

It characterizes exponentially fitted Runge-Kutta methods that maintain conservation laws and applies them to wave equations in physics.

Findings

Methods effectively preserve mass and momentum conservation.

Enhanced accuracy for oscillatory wave solutions.

Comparison shows advantages over existing methods.

Abstract

The exponential fitting technique uses information on the expected behaviour of the solution of a differential problem to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In this cases, compared to standard methods, they have proved to be very accurate even using large integration steps. 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 benchmark problems we consider wave equations arising as models in several fields such as fluid dynamics and quantum physics, and derive exponentially fitted methods that preserve their conservation laws of mass (or charge) and momentum. The proposed methods are applied to approximate breather wave solutions and are compared to other known methods of the same order.