text: Variational time integrators are derived in the context of discrete mechanical systems. In this area, the governing equations for the motion of the mechanical system are built following two steps: (a) Postulating a discrete action; (b) Computing the stationary value for the discrete action. The former is formulated by considering Lagrangian (or Hamiltonian) systems with the discrete action being constructed through numerical approximations of the action integral. The latter derives the discrete Euler-Lagrange equations whose solutions give the variational time integrator. In this paper, we build variational time integrators in the context of smoothed particle hydrodynamics (SPH). So, we start with a variational formulation of SPH for fluids. Then, we apply the generalized midpoint rule, which depends on a parameter @math , in order to generate the discrete action. Then, the step (b) yields a variational time integration scheme that reduces to a known explicit one if @math but it is implicit otherwise. Hence, we design a fixed point iterative method to approximate the solution and prove its convergence condition. Besides, we show that the obtained discrete Euler-Lagrange equations preserve linear momentum. In the experimental results, we consider artificial viscous as well as boundary interaction effects and simulate a dam breaking set up. We compare the explicit and implicit SPH solutions and analyze momentum conservation of the dam breaking simulations.
  @cite_9: This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.
  @cite_16: The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior. This analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d'Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run.
  @cite_12: 
summary: The key elements in variational time integrators are the discrete action sum, the discrete Euler-Lagrange equations and the discrete Noether's theorem that were clearly understood due to early works (see @cite_9 and references therein). Numerical aspects and convergence properties were specifically considered in @cite_16 @cite_12 .