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Error estimates for the standard Galerkin-finite element method for the shallow water equations


Authors: D. C. Antonopoulos and V. A. Dougalis
Journal: Math. Comp. 85 (2016), 1143-1182
MSC (2010): Primary 65M60, 35L60
DOI: https://doi.org/10.1090/mcom3040
Published electronically: September 21, 2015
MathSciNet review: 3454361
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Abstract: We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension and also the analogous problem for a symmetric variant of the system. Assuming smoothness of solutions, we discretize these problems in space using standard Galerkin-finite element methods and prove $ L^{2}$-error estimates for the semidiscrete problems for quasiuniform and uniform meshes. In particular we show that in the case of spatial discretizations with piecewise linear continuous functions on a uniform mesh, suitable compatibility conditions at the boundary and superaccuracy properties of the $ L^{2}$ projection on the finite element subspaces lead to an optimal-order $ O(h^{2})$ $ L^{2}$-error estimate. We also examine the temporal discretization of the semidiscrete problems by a third-order explicit Runge-Kutta method due to Shu and Osher and prove $ L^{2}$-error estimates of optimal order in the temporal variable under a Courant-number stability condition. In a final section of remarks we prove optimal-order $ L^{2}$-error estimates for smooth spline spatial discretizations of the periodic initial-value problem for the systems. We also prove that small-amplitude, appropriately transformed solutions of the symmetric system are close to the corresponding solutions of the usual system while they are both smooth, thus providing a justification of the symmetric system.


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Additional Information

D. C. Antonopoulos
Affiliation: Department of Mathematics, University of Athens, 15784 Zographou, Greece — and — Institute of Applied and Computational Mathematics, FORTH, 70013 Heraklion, Greece
Email: antonod@math.uoa.gr

V. A. Dougalis
Affiliation: Department of Mathematics, University of Athens, 15784 Zographou, Greece — and — Institute of Applied and Computational Mathematics, FORTH, 70013 Heraklion, Greece
Email: doug@math.uoa.gr

DOI: https://doi.org/10.1090/mcom3040
Keywords: Shallow water equations, first-order quasilinear hyperbolic systems, standard Galerkin method, Shu-Osher scheme.
Received by editor(s): March 25, 2014
Received by editor(s) in revised form: November 23, 2014
Published electronically: September 21, 2015
Additional Notes: At http://arxiv.org/abs/1403.5699 interested readers may find an extended version of the present paper (\cite{ad2arxiv}), including additional results as well as details of proofs omitted herein.
Article copyright: © Copyright 2015 American Mathematical Society