×

Analysis of the heterogeneous multiscale method for gas dynamics. (English) Zbl 1177.76355

Summary: 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 earlier, 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.

MSC:

76N15 Gas dynamics (general theory)
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI Euclid