Abstract
In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L2-norm is in both cases of optimal order and proportional to O(Δt2+hp+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.
Similar content being viewed by others
References
D. N. Arnold (1982) ArticleTitleAn interior penalty finite element method with discontinuous elements SIAM J. Numer. Anal. 19 IssueID4 742–760 Occurrence Handle10.1137/0719052
D. N. Arnold F. Brezzi B. Cockburn L. D. Marini (2002) ArticleTitleUnified analysis of discontinuous Galerkin Methods for Elliptic Problems SIAM J. Numer. Anal. 39 IssueID5 1749–1779 Occurrence Handle10.1137/S0036142901384162
F. Bassi S. Rebay G. Mariotti S. Pedinotti M. Savini (1997) A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows R. Decupere G. Dibelius (Eds) Proceedings of the second European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics Antwerpen Belgium 99–108
H. Braess P. Wriggers (2000) ArticleTitleArbitrary Lagrangian Eulerian finite element analysis of free surface flow Comput. Meth. Appl. Mech. Eng. 190 95–109 Occurrence Handle10.1016/S0045-7825(99)00416-8
S. C. Brenner L. R. Scott (Eds) (2002) The Mathematical Theory of Finite Element Methods EditionNumber2 Springer-Verlag New York
Brezzi, F., Manzini, G., Marini, D., Pietra, P., and Russo, A. (1999). Discontinuous finite elements for diffusion problems. In Atti Convegno in Onore di F. Brioschi, Instituto Lombardo, Accademia di Scienze e Lettere, Milan, Italy, pp. 197–217.
F. Brezzi G. Manzini D. Marini P. Pietra A. Russo (2000) ArticleTitleDiscontinuous Galerkin Approximations for Elliptic Problems Numer. Meth. Part. Diff. Eq. 16 IssueID4 365–378 Occurrence Handle10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y
X. Cai H. P. Langtangen B. F. Nielsen A. Tvieto (1998) ArticleTitleA finite element method for fully nonlinear water waves J. Comput. Phys 143 544–568 Occurrence Handle10.1006/jcph.1998.9997
P. Castillo (2002) ArticleTitlePerformance of discontinuous Galerkin methods for elliptic PDES SIAM J. Sci. Comput. 24 IssueID2 524–547 Occurrence Handle10.1137/S1064827501388339
P. Castillo B. Cockburn I. Perugia D. Schotzau (2000) ArticleTitleAn a priori error analysis of the local discontinuous Galerkin method for elliptic problems SIAM J. Numer. Anal. 38 IssueID5 1676–1706 Occurrence Handle10.1137/S0036142900371003
Ciarlet, P. G. (1991). Basic error estimates for elliptic problems. In Ciarlet, P. G., and Lions, J. L. (eds.),Handbook of Numerical Analysis,Vol. II (Part 1), North Holland, Amsterdam.
B. Cockburn C. -W. Shu (1998) ArticleTitleThe local discontinuous Galerkin time-dependent method for convection-diffusion systems SIAM J. Numer. Anal. 35 2440–2463 Occurrence Handle10.1137/S0036142997316712
Dautray, R., and Lions, J. L. (1988).Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer-Verlag.
C. Dawson (2002) ArticleTitleThe \({\cal P}^{k+1}-{\cal S}^k\) local discontinuous Galerkin method for elliptic equations SIAM J. Numer. Anal. 40 IssueID6 2151–2170 Occurrence Handle10.1137/S0036142901397599
C. Dawson J. Proft (2002) ArticleTitleDiscontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations Comput. Meth. Appl. Mech. Eng. 191 IssueID41–42 4721–4746 Occurrence Handle10.1016/S0045-7825(02)00402-4
C. Dawson J. Proft (2003) ArticleTitleDiscontinuous/continuous Galerkin methods for coupling the primitive and wave continuity equations of shallow water Comput. Meth. Appl. Mech. Eng. 192 IssueID47–48 5123–5145 Occurrence Handle10.1016/j.cma.2003.07.004
M. J. Lighthill (Eds) (1978) Waves in Fluids Cambridge University Press Cambridge, UK
J. N. Newman (Eds) (1977) Marine Hydrodynamics MIT Press Cambridge, MA
I. Robertson S. J. Sherwin (1999) ArticleTitleFree-surface flow simulation using hp/Spectral elements J. Comput. Phys. 155 26–53 Occurrence Handle10.1006/jcph.1999.6328
J. J. Sudirham J. J. W. Van der Vegt R. M. J. Van Damme (2003) ArticleTitleA study on discontinuous Galerkin finite element methods for Elliptic problems Tech. Memorandum Dept. Appl. Math. 1690 1–21
Thomée, V. (1997).Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag.
J. H. Westhuis (Eds) (2001) The Numerical Simulation of Nonlinear Waves in a Hydrodynamic Model Test Basin University of Twente, Enschede The Netherlands
G. B. Whitham (Eds) (1974) Linear and Nonlinear Waves Wiley New York
G. X. Wu R. E. Taylor (1994) ArticleTitleFinite element analysis of two-dimensional non-linear transient water waves Appl. Ocean Res. 16 363–372 Occurrence Handle10.1016/0141-1187(94)00029-8
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
van der Vegt, J., Tomar, S. Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves. J Sci Comput 22, 531–567 (2005). https://doi.org/10.1007/s10915-004-4149-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10915-004-4149-1