A new implementation of symplectic Runge–Kutta methods

We propose a new iteration for solving implicit Runge–Kutta equations, using the Jacobian of the vector field and matrix–vector multiplies whose extra cost for certain structured problems is negligible. The iteration reduces the number of vector field evaluations almost to that of Newton’s method, often only 1 or 2 per time step, making symplectic Runge–Kutta methods efficient even at relatively large time steps.