Multi-step numerical methods derived using discrete Lagrangian integrators

On the basis of the variational integrators theory, we initially examine the possibility of deriving multi-step numerical methods. Then, we propose an integration technique that approximates the action integral within one time interval by using appropriate expressions for the relevant configurations and velocities. These approximations depend on a specific number of known configurations defined at previous time nodes. Multi-step numerical methods can finally be deduced, by defining, as usually, the Lagrange function as a weighted sum over the discrete Lagrangians corresponding to each of the curve segments and using the discrete Euler-Lagrange equations.