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Error estimates for Galerkin finite element methods for the Camassa–Holm equation

journal contribution
posted on 17.06.2020 by DC Antonopoulos, VA Dougalis, Dimitrios Mitsotakis
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order L2-error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

History

Preferred citation

Antonopoulos, D.C., Dougalis, V.A. & Mitsotakis, D.E. (2019). Error estimates for Galerkin finite element methods for the Camassa–Holm equation. Numerische Mathematik, 142(4), 833-862. https://doi.org/10.1007/s00211-019-01045-7

Journal title

Numerische Mathematik

Volume

142

Issue

4

Publication date

01/08/2019

Pagination

833-862

Publisher

Springer Science and Business Media LLC

Publication status

Published

Online publication date

11/05/2019

ISSN

0029-599X

eISSN

0945-3245

Language

en

Exports