Explicit computational wave propagation in micro-heterogeneous media

• Published in: BIT Vol. 59; no. 2; pp. 443 - 462
• Main Authors: Maier, Roland, Peterseim, Daniel
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2019; Springer Nature B.V
• Subjects: Acoustic propagation; Computational Mathematics and Numerical Analysis; Elastic waves; Mathematics; Mathematics and Statistics; Numeric Computing; Propagation; Solvers; Stiffness matrix; Wave propagation; Multiscale method; Explicit time stepping; Heterogeneous media; Hyperbolic equation; 35L05; 65M60; Numerical homogenization; 65M12
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-018-0735-8

Explicit time stepping schemes are popular for linear acoustic and elastic wave propagation due to their simple nature which does not require sophisticated solvers for the inversion of the stiffness matrices. However, explicit schemes are only stable if the time step size is bounded by the mesh size in space subject to the so-called CFL condition. In micro-heterogeneous media, this condition is typically prohibitively restrictive because spatial oscillations of the medium need to be resolved by the discretization in space. This paper presents a way to reduce the spatial complexity in such a setting and, hence, to enable a relaxation of the CFL condition. This is done using the Localized orthogonal decomposition method as a tool for numerical homogenization. A complete convergence analysis is presented with appropriate, weak regularity assumptions on the initial data.