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

Bona, J.; Chen, H.; Karakashian, O.; Xing, Y.

doi:10.1090/S0025-5718-2013-02661-0

Conservative, discontinuous Galerkin–methods for the generalized Korteweg–de Vries equation
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by J. L. Bona, H. Chen, O. Karakashian and Y. Xing

Math. Comp. 82 (2013), 1401-1432

DOI: https://doi.org/10.1090/S0025-5718-2013-02661-0

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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