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A Posteriori Error Estimates for Conservative Local Discontinuous GalerkinMethods for the Generalized Korteweg-de Vries Equation

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:  22 June 2016

Ohannes Karakashian  and
Yulong Xing
Ohannes Karakashian*
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA
Yulong Xing*
Affiliation:
Department of Mathematics, University of California Riverside, Riverside, CA 92521, USA
*
*Corresponding author. Email addresses:ohannes@math.utk.edu (O. Karakashian), xingy@ucr.edu (Y. Xing)
*Corresponding author. Email addresses:ohannes@math.utk.edu (O. Karakashian), xingy@ucr.edu (Y. Xing)
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Abstract

We construct and analyze conservative local discontinuous Galerkin (LDG) methods for the Generalized Korteweg-de-Vries equation. LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives. The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution, and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction. Numerical experiments are provided to verify the theoretical estimates.

Keywords

Discontinuous Galerkin methodsKorteweg-de-Vries equationa posteriori error estimateconservative methods

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 250 - 278
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Adams, R.. Sobolev Spaces. Academic Press, New York, 1975.Google Scholar
[2] Akrivis, G., Makridakis, C., and Nochetto, R. H.. A posteriori error estimates for the Crank–Nicolson method for parabolic equations. Math. Comp., 75:511–531, 2006.Google Scholar
[3] Angulo, J., Bona, J.L., Linares, F., and Scialom, M.. Scaling, stability and singularities for nonlinear dispersive wave equations: The critical case. Nonlinearity, 15:759–786, 2002.Google Scholar
[4] Baker, G., Dougalis, V.A., and Karakashian, O.A.. Convergence of Galerkin approximations for the Korteweg-de Vries equation. Math. Comp., 40:419–433, 1983.Google Scholar
[5] Baker, G., Jureidini, W., and Karakashian, O.A.. Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Num. Anal., 27:1466–1485, 1990.Google Scholar
[6] Bassi, F. and Rebay, S.. A high-order accurate discontinuous finite element method for the numerical solution of the compressibleNavier-Stokes equations. J. of Comput. Phys., 131:267–279, 1997.CrossRefGoogle Scholar
[7] Benjamin, T.B.. The stability of solitary waves. Proc. Royal Soc. London, Ser. A, 328:153–183, 1972.Google Scholar
[8] Bona, J.L.. On the stabilty theory of solitary waves. Proc. Royal Soc. London, Ser. A, 349:363–374, 1975.Google Scholar
[9] Bona, J.L., Chen, H., Karakashian, O., and Xing, Y.. Conservative discontiuous Galerkin methods for generalized Korteweg-de Vries equation. Math. Comp., 82(283):1401–1432, 2013.Google Scholar
[10] Brenner, S. and Scott, L.R.. The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer Verlag, New York, third edition, 2002.Google Scholar
[11] Cheng, Y. and Shu, C.-W.. A discontinuous finite element method for time dependent partial differential equations with higher order derivatives. Math. Comp., 77:699–730, 2008.Google Scholar
[12] Cheng, Y. and Shu, C.-W.. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J. of Comput. Phys., 227:9612–9627, 2008.CrossRefGoogle Scholar
[13] Cockburn, B., Hou, S., and Shu, C.-W.. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comp., 54:545–581, 1990.Google Scholar
[14] Cockburn, B., Lin, S.-Y., and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. of Comput. Phys., 84:90–113, 1989.Google Scholar
[15] Cockburn, B. and Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comp., 52:411–435, 1989.Google Scholar
[16] Cockburn, B. and Shu, C.-W.. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal., 35:2440–2463, 1998.Google Scholar
[17] Cockburn, B. and Shu, C.-W.. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. of Comput. Phys., 141:199–224, 1998.Google Scholar
[18] Cockburn, B. and Shu, C.-W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16:173–261, 2001.CrossRefGoogle Scholar
[19] Dougalis, V.A. and Karakashian, O.A.. On some high order accurate fully discrete Galerkin methods for the Kortweg-de Vries equation. Math. Comp., 45:329–345, 1985.Google Scholar
[20] Hufford, C. and Xing, Y.. Superconvergence of the local discontinuous galerkin method for the linearized korteweg-de vries equation. J. Comput. Appl. Math., 255:441–455, 2014.Google Scholar
[21] Karakashian, O. and Makridakis, C.. A posteriori error estimates for discontinuous galerkin methods for the generalized korteweg-de vries equation. Math. Comp., 84(293):1145–1167, 2015.Google Scholar
[22] Karakashian, O. and Xing, Y.. On the construction of compatible and optimal-order initial approximations for local discontinuous Galerkin methods. In preparation.Google Scholar
[23] Martel, Y. and Merle, F.. Stability of blow-up profile and lower bounds on the blow up rate for the critical generalized KdV equation. Annals Math., 155:235–280, 2002.Google Scholar
[24] Merle, F.. Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. American Math. Soc., 14:666–678, 2001.Google Scholar
[25] Reed, W.H. and Hill, T.R.. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[26] Shu, C.-W.. Discontinuous Galerkin methods for time-dependent problems: Survey and recent developments. In Feng, X., Karakashian, O., and Xing, Y., editors, Proceedings of the 2012 John H. Barrett Memorial Lectures: Recent developments in discontinuous Galerkin finite element methods for partial differential equation, volume 157 of IMA volumes in Mathematics and its applications, pages 25–62. Springer, 2014.Google Scholar
[27] Xu, Y. and Shu, C.-W.. Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Computer Methods in Appl. Mech. and Eng., 196:3805–3822, 2007.Google Scholar
[28] Xu, Y. and Shu, C.-W.. Optimal error estimates of the semi-discrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal., 50:79–104, 2012.CrossRefGoogle Scholar
[29] Yan, J. and Shu, C.-W.. A local discontinuous Galerkinmethod for KdV type equations. SIAM J. Numer. Anal., 40:769–791, 2002.Google Scholar