Modified Equations for Variational Integrators
Provides a theoretical foundation for variational integrators by linking discrete and continuous Lagrangians, benefiting numerical analysts and researchers in geometric numerical integration.
The authors present a technique to construct a Lagrangian for the modified equation from the discrete Lagrangian of a variational integrator, showing that the modified equation is Lagrangian. This extends the known Hamiltonian property of symplectic integrators to the variational framework.
It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational side. We present a technique to construct a Lagrangian for the modified equation from the discrete Lagrangian of a variational integrator.