Stochastic PDEs of Fluctuating Hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. We develop and analyze a Multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. In particular, we prove rigorously and demonstrate numerically that our MLMC scheme provides a significant speed-up in the simulation of the Dean–Kawasaki model. Specifically, we quantify how the speed-up factor of suitable MLMC schemes (with respect to a standard Monte Carlo method) increases as the local particle density increases, and show that sizeable speed-ups can be obtained even in regimes of low particle density. Due to the highly singular nature of the Dean–Kawasaki equation caused by the presence of a white noise drift term, the convergence of numerical solutions in strong spatial norms is rather slow for coarse discretizations and fails for too fine discretizations. Instead, our results are formulated entirely in terms of the law of distributions.
Numerical simulations will be presented in the two-dimensional case.
Based on a joint work with Julian Fischer.