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Multiple‐scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations
Author(s) -
Kevorkian J.,
Bosley D. L.
Publication year - 1998
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00088
Subject(s) - conservation law , homogenization (climate) , nonlinear system , mathematics , discontinuity (linguistics) , initial value problem , mathematical analysis , statistical physics , spatial ecology , physics , biodiversity , ecology , quantum mechanics , biology
We consider hyperbolic conservation laws with rapid periodic spatial fluctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are computed asymptotically using multiple spatial and temporal scales to capture the homogenized solution as well as its long‐term behavior. We show that the linear problem may be destabilized through interactions between two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation, introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the homogenized leading‐order solution are more general than their counterparts for conservation laws having no rapid spatial variations. In particular, these equations may be diffusive for certain general flux vectors. Selected examples are solved numerically to substantiate the asymptotic results.