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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation


Authors: J. L. Bona, H. Chen, O. Karakashian and Y. Xing
Journal: Math. Comp. 82 (2013), 1401-1432
MSC (2010): Primary 65N12, 65N30, 35Q35, 35Q51, 35Q53, 35Q86, 76B15, 76B25
Published electronically: January 7, 2013
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Abstract: We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg-de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the $ L^2$-norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling-wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.


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Additional Information

J. L. Bona
Affiliation: Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, Chicago, Illinois 60607
Email: bona@math.uic.edu

H. Chen
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: hchen1@memphis.edu

O. Karakashian
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
Email: ohannes@math.utk.edu

Y. Xing
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996 – and – the Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Email: xingy@math.utk.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02661-0
PII: S 0025-5718(2013)02661-0
Keywords: Discontinuous Galerkin methods, Korteweg–de Vries equation, error estimates, conservation laws
Received by editor(s): June 7, 2011
Received by editor(s) in revised form: December 6, 2011
Published electronically: January 7, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.