Chapter

Numerical Analysis of Multiscale Computations

Volume 82 of the series Lecture Notes in Computational Science and Engineering pp 167-186

Date:

Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time

  • Björn EngquistAffiliated withDepartment of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin
  • , Henrik HolstAffiliated withDepartment of Numerical Analysis, CSC, KTH
  • , Olof RunborgAffiliated withDepartment of Numerical Analysis, CSC and Swedish e-Science Research Center (SeRC) KTH Email author 

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Abstract

Multiscale 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 multiscale wave propagation in the framework of the heterogeneous multiscale method (HMM). The numerical methods couple simulations on macro- and microscales for problems with rapidly oscillating coefficients. The complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the smallest scale, when computing solutions at a fixed time and accuracy. We show numerical examples of the HMM applied to long time integration of wave propagation problems in both periodic and non-periodic medium. In both cases our HMM accurately captures the dispersive effects that occur. We also give a stability proof for the HMM, when it is applied to long time wave propagation problems.