Solving Boundary-value Problems for Systems of Hyperbolic Conservation Laws with Rapidly Varying Coefficients
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.
Yong, DH, Kevorkian, J. Solving boundary-value problems for systems of hyperbolic conservation laws with rapidly varying coefficients. Stud Appl Math. 2002;108(3): 259–303. .