does anyone have experience with symplectic integrators when applied to non-separable Hamiltonians? More specifically with regard to constructing high order symplectic integrators for non-separable Hamiltonians?
I know there is Yoshida's article but this is specific to separable Hamiltonians.
Yoshida's trick applies to all symmetric integrators. So if symplectic structure preservation is important to you then using e.g. the implicit midpoint and Yoshida's trick allows you to make arbitrary high order symplectic methods. Do note however, that Yoshida's trick gives terrible error constants and using e.g. high order Gauss methods (also symplectic) gives much better results. Recent results by Murua et al as well as results by Hairer show how fast fixed point iterations can be used to solve the resulting implicit/nonlinear equations when using Gauss methods.