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Variational time integrators are derived in the context of discretemechanical systems. In this area, the governing equations for the motion of themechanical system are built following two steps: (a) Postulating a discreteaction; (b) Computing the stationary value for the discrete action. The formeris formulated by considering Lagrangian (or Hamiltonian) systems with thediscrete action being constructed through numerical approximations of theaction integral. The latter derives the discrete Euler-Lagrange equations whosesolutions give the variational time integrator. In this paper, we buildvariational time integrators in the context of smoothed particle hydrodynamics(SPH). So, we start with a variational formulation of SPH for fluids. Then, weapply the generalized midpoint rule, which depends on a parameter $\alpha$, inorder to generate the discrete action. Then, the step (b) yields a variationaltime integration scheme that reduces to a known explicit one if$\alpha\in\{0,1\}$ but it is implicit otherwise. Hence, we design a fixed pointiterative method to approximate the solution and prove its convergencecondition. Besides, we show that the obtained discrete Euler-Lagrange equationspreserve linear momentum. In the experimental results, we consider artificialviscous as well as boundary interaction effects and simulate a dam breaking setup. We compare the explicit and implicit SPH solutions and analyze momentumconservation of the dam breaking simulations.

Variational Time Integration Approach for Smoothed Particle Hydrodynamics Simulation of Fluids