I have to solve equations of motion of a charged particle under the effect of electromagnetic field. Since I have to deal with speed over precision I could not use adaptive stepsize algorithms (like Runge-Kutta Cash-Karp) because they would take too much time. I was looking for an algorithm which is both symplectic (like Boris integration) and exponentially fitted (in order to solve the equation of motion even if the equation is stiff). I found a method but it is for second order differential equations:
https://www.math.purdue.edu/~xiaj/work/SEFRKN.pdf
Later I found a paper which would describe a fourth order symplectic exponentially-fitted Runge-Kutta:
http://users.ugent.be/~gvdbergh/files/publatex/annals1.pdf
Since I have to deal with speed I was looking for a lower order algorithm. Does a 2nd order symplectic exponentially fitted ODE algorithm exist?
