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Solving Boundary‐Value Problems for Systems of Hyperbolic Conservation Laws with Rapidly Varying Coefficients
Author(s) -
Yong Darryl H.,
Kevorkian J.
Publication year - 2002
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 - 10.1111/1467-9590.01412
Subject(s) - conservation law , mathematics , boundary value problem , nonlinear system , boundary (topology) , scale (ratio) , mathematical analysis , consistency (knowledge bases) , laplace transform , hyperbolic partial differential equation , set (abstract data type) , mixed boundary condition , partial differential equation , computer science , geometry , physics , quantum mechanics , programming language
We study how boundary conditions affect the multiple‐scale analysis of hyperbolic conservation laws with rapid spatial fluctuations. The most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions. We show how to overcome this missing boundary condition difficulty for both linear and nonlinear problems through the recovery of boundary information. We introduce two methods for this recovery (multiple‐scale analysis with a reduced set of scales, and a combination of Laplace transforms and multiple scales) and show that they are roughly equivalent. We also show that the recovered boundary information is likely to contain secular terms if the initial conditions are nonzero. However, for the linear problem, we demonstrate how to avoid these secular terms to construct a solution that is valid for all time. For nonlinear problems, we argue that physically relevant problems do not exhibit the missing boundary condition difficulty.