Volume 52, Number 3, May–June 2018
|Page(s)||1195 - 1218|
|Published online||14 September 2018|
Generalized wave propagation problems and discrete exterior calculus★
University of Jyvaskyla, Faculty of Information Technology,
P.O. Box 35,
* Corresponding author: firstname.lastname@example.org
** Present address: University of Helsinki, Department of Physics, P.O. Box 64, 00014 Helsinki, Finland
Accepted: 9 February 2018
We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a “discrete Hodge” adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude.
Mathematics Subject Classification: 35L65 / 58A10 / 58A14 / 58J32 / 58J90 / 65M06 / 65M12 / 65M22
Key words: Differential geometry / exterior algebra / boundary value problems / acoustics / electromagnetism / elasticity / quantum mechanics / finite difference / discrete exterior calculus
The computations presented were performed using the computing resources provided by the CSC – IT center for science, owned by the Finnish Ministry of Education and Culture. The research was supported by the ERC Advanced Grant project SAEMPL (No. 320733). We also acknowledge the anonymous referees for valuable comments, which significantly improved the presentation of the paper.
© EDP Sciences, SMAI 2018
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