Cheap implicit symplectic integrators Original Research Article

The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems by, say, the leapfrog method the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic methods like implicit midpoint have less severe stability restrictions but the cost of solving large nonlinear systems with dense Jacobian matrices is probably too high to make them worthwhile. To bring down the cost of implicit methods, we have designed (i) mixed implicit-explicit, and (ii) linearly implicit methods that retain the property of being symplectic.