On the infinite particle limit in Lagrangian dynamics and convergence of optimal transportation meshfree methods
Provides theoretical foundations for meshfree methods in computational physics, but is primarily a mathematical analysis without immediate practical impact.
The paper proves Gamma-convergence of discrete action functionals for Lagrangian systems with infinitely many particles to a continuum limit, and establishes convergence of stationary points. This provides a convergence analysis for optimal transportation meshfree methods approximating particle flows.
We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Gamma-converge to a continuum action functional acting on probability measures of particle trajectories. Also the convergence of stationary points of the action is established. Minimizers of the limiting functional and, more generally, limiting distributions of stationary points are investigated and shown to be concentrated on orbits of the Euler-Lagrange flow. We also consider time discretized systems. These results in particular provide a convergence analysis for optimal transportation meshfree methods for the approximation of particle flows by finite discrete Lagrangian dynamics.