Volume preserving diffeomorphisms and the Kahan method

Lu Li (Norwegian university of Science and Technology), Elena Celledoni

Kahan’s method is a special numerical integration method that works very well for certain quadratic differential equations. A modified measure and one or several modified integrals are preserved by this method for some special classes of quadratic vector fields. In this talk, we apply Kahan’s method to a general vector filed \(F\) and give a general condition on its Jacobian matrix \(F'\) guaranteeing that a modified measure is preserved. Connections to the discretization of the group of volume preserving diffeomorphisms arising in certain classes of partial differential equations will be discussed.