A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations

Shaoqiang Tang, Thomas Y. Hou*, Wing Kam Liu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Scopus citations

Abstract

In this paper, we propose a pseudo-spectral multiscale method for simulating complex systems with more than one spatial scale. Using a spectral decomposition, we split the displacement into its mean and fluctuation parts. Under the assumption of localized nonlinear fluctuations, we separate the domain into an MD (Molecular Dynamics) subdomain and an MC (MacrosCopic) subdomain. An interfacial condition is proposed across the two scales, in terms of a time history treatment. In the special case of a linear system, this is the first exact interfacial condition for multiscale computations. Meanwhile, we design coarse grid equations using a matching differential operator approach. The coarse grid discretization is of spectral accuracy. We do not use a handshaking region in this method. Instead, we define a coarse grid over the whole domain and reassign the coarse grid displacement in the MD subdomain with an average of the MD solution. To reduce the computational cost, we compute the dynamics of the coarse grid displacement and relate it to the mean displacement. Our method is therefore called a pseudo-spectral multiscale method. It allows one to reach high resolution by balancing the accuracy at the coarse grid with that at the interface. Numerical experiments in one- and two-space dimensions are presented to demonstrate the accuracy and the robustness of the method.

Original languageEnglish (US)
Pages (from-to)57-85
Number of pages29
JournalJournal of Computational Physics
Volume213
Issue number1
DOIs
StatePublished - Mar 20 2006

Funding

We thank Dr. Eduard Karpov, Dr. Harold Park, Mr. Justin Mach, Ms. Sydney Garstang and Mr. Wuan Luo for stimulating discussions. This research is partially supported by an NSF FRG Grant DMS-0353838 and an NSF ITR Grant ACI-0204932. S.T. acknowledge partial support of Chinese Special Funds for Major State Basic Research Project Nonlinear Sciences and NSFC under Grant Nos. 90407021 and 10271003. W.K.L. gratefully acknowledge the support of the NSF, and also the NSF-IGERT program. We thank the NSF Summer Institute of Nano Mechanics and Materials, and the Army Research Office (ARO) for their support of this research.

Keywords

  • Coarse grid equations
  • Interfacial conditions
  • Multiscale method

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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