V.A. Marchenko and E.Y. Khruslov: Homogenization of Partial Differential Equations
This book, which was first published in Russian in 2005, deals with the homogenization of partial differential equations (pdes) of elliptic and parabolic types. The authors study both boundary value problems posed in highly perforated domains or equations with rapidly oscillating coefficients. The standard theory of homogenization (periodic theory, uniformly elliptic oscillating coefficients) is supposed to be already known and the emphasis is put on nonstandard situations leading to multicomponent or nonlocal equations.
Abstract The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems by, say, the leapfrog method the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic methods like implicit midpoint have less severe stability restrictions but the cost of solving large nonlinear systems with dense Jacobian matrices is probably too high to make them worthwhile. To bring down the cost of implicit methods, we have designed (i) mixed implicit-explicit, and (ii) linearly implicit methods that retain the property of being symplectic.