Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T04:00:03.312Z Has data issue: false hasContentIssue false
  • English
  • Français

FOURTH-ORDER NUMERICAL METHODS FOR THE COUPLED KORTEWEG–DE VRIES EQUATIONS

Part of: Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  16 March 2015

I. A. KOROSTIL  and
S. R. CLARKE
I. A. KOROSTIL*
Affiliation:
The Kirby Institute, University of New South Wales, Sydney, Australia email [email protected]
S. R. CLARKE
Affiliation:
School of Mathematical Sciences, Monash University, Melbourne, Australia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compare six fixed-stepsize fourth-order numerical methods for a number of test problems described by a system of coupled Korteweg–de Vries equations. Particular attention is paid to the ability of these methods to preserve fixed points (solitary waves) and the invariants of the system, and establishing to what extent the conservation of integral invariants is indicative of the solution error for these methods.

Keywords

coupled Korteweg–de Vries equationsnumerical methodsexponential integratorsimplicit Runge–Kutta method

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries)
Secondary: 65M70: Spectral, collocation and related methods
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 3 , January 2015 , pp. 275 - 285
Copyright
© 2015 Australian Mathematical Society 

References

Ashi, H. A., Cummings, L. J. and Matthews, P. C., “Comparison of methods for evaluating functions of a matrix exponential”, Appl. Numer. Math. 59 (2009) 468–486 ;doi:10.1016/j.apnum.2008.03.039.CrossRefGoogle Scholar
Calvo, M. P., De Frutos, J. and Novo, J., “Linearly implicit Runge–Kutta methods for advection-reaction-diffusion equations”, Appl. Numer. Math. 37 (2001) 535–549 ;doi:10.1016/S0168-9274(00)00061-1.CrossRefGoogle Scholar
Celledoni, E., Marthinsen, A. and Owren, B., “Commutator-free Lie group methods”, Future Gener. Comput. Syst. 19 (2003) 341–352; doi:10.1016/S0167-739X(02)00161-9.CrossRefGoogle Scholar
Cox, S. M. and Matthews, P. C., “Exponential time differencing for stiff systems”, J. Comput. Phys. 176 (2002) 430–455; doi:10.1006/jcph.2002.6995.CrossRefGoogle Scholar
Eckart, C., “Internal waves in the ocean”, Phys. Fluids 4 (1961) 791; doi:10.1063/1.1706408.CrossRefGoogle Scholar
García-Archilla, B., “Some practical experience with the time integration of dissipative equations”, J. Comput. Phys. 122 (1995) 25–29; doi:10.1006/jcph.1995.1193.CrossRefGoogle Scholar
Gear, J. A. and Grimshaw, R., “Weak and strong interactions between internal solitary waves”, Stud. Appl. Math. 70 (1984) 235–258.CrossRefGoogle Scholar
Gottwald, G. and Grimshaw, R., “The formation of coherent structures in the context of blocking”, J. Atmos. Sci. 56 (1999) 3640–3662; doi:10.1175/1520-0469(1999)056<3640:TFOCSI>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Grimshaw, R. and Skyrnnikov, Y., “Long-wave instability in a three-layer stratified shear flow”, Stud. Appl. Math. 108 (2002) 77–88; doi:10.1111/1467-9590.01424.CrossRefGoogle Scholar
Kassam, A.-K. and Trefethen, L. N., “Fourth-order time-stepping for stiff PDEs”, SIAM J. Sci. Comput. 26 (2005) 1214; doi:10.1137/S1064827502410633.CrossRefGoogle Scholar
Klein, C., “Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation”, Electron. Trans. Numer. Anal. 29 (2008) 116–135; http://www.emis.de/journals/ETNA/vol.29.2007-2008/pp116-135.dir/pp116-135.pdf.Google Scholar
Krogstad, S., “Generalized integrating factor methods for stiff PDEs”, J. Comput. Phys. 203 (2005) 72–88; doi:10.1016/j.jcp.2004.08.006.CrossRefGoogle Scholar
Lawson, J. D., “Generalized Runge–Kutta processes for stable systems with large Lipschitz constants”, SIAM J. Numer. Anal. 4 (1967) 372–380; doi:10.1137/0704033.CrossRefGoogle Scholar
Minchev, B., “Integrating factor methods as exponential integrators”, Large-Scale Sci. Comput. (2006) 380–386; doi:10.1007/1166680643.CrossRefGoogle Scholar
Minchev, B. and Wright, W., “A review of exponential integrators for first order semi-linear problems”, Preprint, 2/2005, Department of Mathematics, Norwegian University of Science and Technology, Trondheim, Norway. http://cds.cern.ch/record/848122/files/cer-002531456.pdf.Google Scholar
Mitsudera, H., “Eady solitary waves: A theory of type B cyclogenesis”, J. Atmos. Sci. 51 (1994) 3137–3154; doi:10.1175/1520-0469(1994)0513C3137:ESWATO3E2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Nouri, F. Z. and Sloan., D. M., “A comparison of Fourier pseudospectral methods for the solution of the Korteweg-de Vries equation”, J. Comput. Phys. 83 (1989) 324–344 ;doi:10.1016/0021-9991(89)90122-8.CrossRefGoogle Scholar
Ostermann, A., Thalhammer, M. and Wright, W. M., “A class of explicit exponential general linear methods”, BIT Numer. Math. 46 (2006) 409–431; doi:10.1007/s10543-006-0054-3.CrossRefGoogle Scholar
Skaflestad, B. and Wright, W. M., “The scaling and modified squaring method for matrix functions related to the exponential”, Appl. Numer. Math. 59 (2009) 783–799 ;doi:10.1016/j.apnum.2008.03.035.CrossRefGoogle Scholar
Skrynnikov, Y., “Nonlinear coupled waves in stratified flows”, Ph.D. Thesis, Monash University, Melbourne, 2002.Google Scholar
Taha, T. R. and Ablowitz, M. J., “Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation”, J. Comput. Phys. 55 (1984) 231–253; doi:10.1016/0021-9991(84)90004-4.CrossRefGoogle Scholar
Wright, J. D. and Scheel, A., “Solitary waves and their linear stability in weakly coupled KdV equations”, Z. Angew. Math. Phys. 58 (2007) 535–570; doi:10.1007/s00033-006-6076-5.CrossRefGoogle Scholar
Yang, J., Nonlinear Waves in Integrable and Nonintegrable Systems (Society for Industrial Mathematics, Philadelphia, 2010).CrossRefGoogle Scholar