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Error Estimates for well-balanced and time-split schemes on a locally damped wave equation


Authors: Debora Amadori and Laurent Gosse
Journal: Math. Comp. 85 (2016), 601-633
MSC (2010): Primary 35L71, 65M15, 74J20
DOI: https://doi.org/10.1090/mcom/3043
Published electronically: October 16, 2015
MathSciNet review: 3434873
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Abstract: A posteriori $ L^1$ error estimates are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary $ 3 \times 3$ system which linear convective structure allows to reduce the Godunov scheme with optimal Courant number (corresponding to $ \Delta t=\Delta x $) to a wavefront-tracking algorithm free from any step of projection onto piecewise constant functions. A fundamental difference in the total variation estimates is proved, which partly explains the discrepancy of the FS method when the dissipative (sink) term displays an explicit dependence in the space variable. Numerical tests are performed by means of stationary exact solutions of the linear damped wave equation.


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Additional Information

Debora Amadori
Affiliation: DISIM, Università degli Studi dell’Aquila, 67100 L’Aquila, Italy
Email: amadori@univaq.it

Laurent Gosse
Affiliation: IAC–CNR “Mauro Picone” (sezione di Roma) - Via dei Taurini, 19 - 00185 Rome, Italy
Email: l.gosse@ba.iac.cnr.it

DOI: https://doi.org/10.1090/mcom/3043
Received by editor(s): August 25, 2013
Received by editor(s) in revised form: September 1, 2013, March 16, 2014, and September 30, 2014
Published electronically: October 16, 2015
Article copyright: © Copyright 2015 American Mathematical Society