In this paper, we develop an algorithm refinement (AR) scheme for an excluded random walk model whose mean field behavior is given by the viscous Burgers' equation. AR hybrids use the adaptive mesh refinement framework to model a system using a molecular algorithm where desired while allowing a computationally faster continuum representation to be used in the remainder of the domain. The focus in this paper is the role of fluctuations on the dynamics. In particular, we demonstrate that it is necessary to include a stochastic forcing term in Burgers' equation to accurately capture the correct behavior of the system. The conclusion we draw from this study is that the fidelity of multiscale methods that couple disparate algorithms depends on the consistent modeling of fluctuations in each algorithm and on a coupling, such as algorithm refinement, that preserves this consistency.
Bibliographical noteFunding Information:
The authors thank Berni Alder and Frank Alexander for helpful discussions. This work was supported by the Applied Mathematical Sciences Program of the DOE Office of Mathematics, Information, and Computational Sciences, Under Contract DE-AC03-76SF00098 as well as the DOE Computational Science Graduate Fellowship, Under Grant Number DE-FG02-97ER25308.
- Adaptive mesh refinement
- Algorithm refinement
- Asymmetric excluded random walk
- Burgers' equation
- Hybrid methods
- Stochastic partial differential equations