TY - JOUR
T1 - THE HETEROGENEOUS MULTISCALE METHODS
AU - Weinan, E.
AU - Engquist, Bjorn
N1 - Funding Information:
We are grateful for many inspiring discussions with Yannis Kevrekidis in which he has outlined his program of macroscale analysis based on microscale solvers. We are also grateful to Eric Vanden-Eijnden and Olof Runborg for stimulating discussions and to Assyr Abdulle and Chris Schwab for suggestions that improved the first draft of the paper. The work of E is supported in part by an ONR grant N00014-01-1-0674. The work of Engquist is supported in part by NSF grant DMS-9973341
Publisher Copyright:
© 2003 International Press
PY - 2003
Y1 - 2003
N2 - The heterogeneous multiscale method (HMM) is presented as a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids. Both variational and dynamic problems are considered. The method relies on an efficient coupling between the macroscopic and microscopic models. In cases when the macroscopic model is not explicitly available or invalid, the microscopic solver is used to supply the necessary data for the macroscopic model. Scale separation can be exploited to considerably reduce the complexity of the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [13], quasicontinuum methods [73, 69, 68] and projective methods for systems with multiscales [34, 35], HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed
AB - The heterogeneous multiscale method (HMM) is presented as a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids. Both variational and dynamic problems are considered. The method relies on an efficient coupling between the macroscopic and microscopic models. In cases when the macroscopic model is not explicitly available or invalid, the microscopic solver is used to supply the necessary data for the macroscopic model. Scale separation can be exploited to considerably reduce the complexity of the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [13], quasicontinuum methods [73, 69, 68] and projective methods for systems with multiscales [34, 35], HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed
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U2 - 10.4310/CMS.2003.v1.n1.a8
DO - 10.4310/CMS.2003.v1.n1.a8
M3 - Article
AN - SCOPUS:85128805678
SN - 1539-6746
VL - 1
SP - 87
EP - 132
JO - Communications in Mathematical Sciences
JF - Communications in Mathematical Sciences
IS - 1
ER -