Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2025-01-31 DOI:10.1137/23m1617345

Federico Cornalba, Julian Fischer

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 262-287, February 2025.
Abstract. Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.