Methods and Applications of Analysis

Volume 11 (2004)

Number 4

Analysis of the heterogeneous multiscale method for gas dynamics

Pages: 557 – 572

DOI: http://dx.doi.org/10.4310/MAA.2004.v11.n4.a7

Authors

Weinan E

Xiantao Li

Abstract

We carry out an error analysis for the heterogeneous multi-scale method for the case when the macroscale process is that of gas dynamics or more generally nonlinear conservation laws and the microscale process is an atomistic model such as kinetic Monte Carlo methods or molecular dynamics (MD). We will consider problems of type B as defined in [4], i.e. the macroscale constitutive relations are unknown and are extracted from the microscopic model. In addition to the standard error in the macroscale solver, a new error term occurs in estimating the data, here the fluxes. This new error term consists of three parts: the relaxation error, the sampling error and the error due to the finite size of the atomistic simulation. Our results serve as guidelines for designing multiscale methods, as was done in [13, 16].

2010 Mathematics Subject Classification

Primary 76Mxx. Secondary 35L65, 65Mxx, 76Nxx.

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