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Center for Nonlinear Studies |
We are interested in a numerical solution of the acoustic wave equation (in the time domain) that is efficient and has small long-time integration error. There exists a large number of numerical schemes for solution of wave equations which are asymptotically accurate in the limit of infinite resolution. Unfortunately, in practice one always operates with finite resolution. As a consequence the numerical schemes may exhibit numerical anomalies – behavior that is not observed in the physical problem. For the wave equation the typical anomalies, and the main source of long integration error, are numerical dispersion and anisotropy – the phenomena predicting different speed of wave propagation depending on their wavelength and orientation. In contrast, in the underlying physical problem the propagation speed is constant for waves with all wavelengths moving in all directions. If the numerical dispersion is strong enough or the integration times are long enough, the wave profiles predicted by the numerical solutions may have very little resemblance with the physical ones. We present an adaptation technique, based on the Mimetic Finite Difference (MFD) discretizations, that allows to minimize the numerical anomalies such as numerical dispersion and anisotropy. The numerical experiments show that the proposed method is consistently better than the classical methods for reducing long-time integration error. The typical L2-error is an order of magnitude smaller for the proposed method. No prior knowledge of MFD methods is required. Familiarity with the classical Finite Element methods will be a plus, but is not a requirement either.
Host: Humberto C Godinez Vazquez, Mathematical Modeling and Analysis Theoretical Division, 5-9188