Communications in Mathematical Sciences

Volume 9 (2011)

Number 1

Multi-scale methods for wave propagation in heterogeneous media

Pages: 33 – 56



Björn Engquist (Department of Mathematics and Institute for Computational Engineering and Sciences, University of Texas at Austin)

Henrik Holst (Department of Numerical Analysis, KTH, Stockholm, Sweden)

Olof Runborg (Department of Numerical Analysis, KTH, Stockholm, Sweden)


Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couple simulations on macro- and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two, and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.


multi-scale, wave propagation, HMM, homogenization

2010 Mathematics Subject Classification

35B27, 35L05, 65N06

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