Fast weak-KAM integrators for separable Hamiltonian systems
This work provides a convergent and efficient numerical method for solving Hamilton-Jacobi equations, which is important for researchers in computational dynamics and optimal control.
The authors propose a numerical scheme for Hamilton-Jacobi equations based on discretizing the Lax-Oleinik semi-group, prove its convergence with error estimates, and show it is a geometric integrator satisfying a discrete weak-KAM theorem. They also demonstrate an efficient implementation using a fast algorithm for min-plus convolutions.
We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.