A Simple Explicit Operator-Splitting Method for Effective Hamiltonians

Understanding effective Hamiltonians quantitatively is essential for the homogenization of Hamilton--Jacobi equations. We propose in this article a simple efficient operator-splitting method for computing effective Hamiltonians when the Hamiltonian is either convex or nonconvex in the gradient variable. To speed up our Lie scheme--based operator-splitting method, we further propose a cascadic initialization strategy so that the steady state of the underlying time-dependent Hamilton--Jacobi equation can be reached more rapidly. Extensive numerical examples demonstrate the efficiency and accuracy of the new algorithm.