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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2018

26-05-2017

High Order \(C^0\)-Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and Application to the Korteweg-de Vries Model

Authors: Sebastian Minjeaud, Richard Pasquetti

Published in: Journal of Scientific Computing | Issue 1/2018

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Abstract

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that it is possible to develop reliable and effective schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) \(H^1\)-conformal Galerkin approximation, namely the Spectral Element Method. The proposed approach is a priori easily extensible to other partial differential equations and to multidimensional problems.
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Appendix
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Literature
1.
Aksan, E.N., Özdes, A.: Numerical solution of the Korteweg-de Vries equation by Galerkin B-spline finite element method. Appl. Math. Comput. 175, 1256–1265 (2006)MathSciNetMATH
2.
Ascher, U.M., Ruth, S.J., Wetton, B.: Implicit–explicit methods for time-dependent PDE’s. SIAM J. Numer. Anal. 32, 797–823 (1995)MathSciNetCrossRefMATH
3.
Ascher, U.M., Ruth, S.J., Spiteri, R.J.: Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Num. Math. 25, 151–167 (1997)MathSciNetCrossRefMATH
4.
Baker, G.A., Dougalis, V.A., Karakashian, O.A.: Convergence of Galerkin approximations for the Korteweg-de Vries equation. Math. Comput. 40(163), 419–433 (1983)MathSciNetCrossRefMATH
5.
Berezin, Y.U., Karpman, V.I.: Nonlinear evolution of disturbances in plasmas and other dispersive media. Sov. Phys. Jetp 24(5), 1049–1055 (1967)
6.
Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Conservative high-order numerical schemes for the generalized Korteweg-de Vries equation. Phil. Trans. R. Soc. Lond. 351, 107–164 (1995)MathSciNetCrossRefMATH
7.
Bona, J.L., Chen, H., Karakashian, O., Xing, Y.: Conservative, discontinuous Galerkin methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)MathSciNetCrossRefMATH
8.
Bueno, N.D., Mastroserio, C.: Explicit methods based on a class of four stage fourth order Runge–Kutta methods for preserving quadratic invariants. J. of Comput. Appl. Math. 20, 247–260 (1996)
9.
Butcher, J.C.: A histoty of Runge–Kutta methods. Appl. Num. Math. 25, 151–167 (1997)CrossRef
10.
Calvo, M., Hernández-Abreu, D., Montijano, J.I., Rández, L.: On the preservation of invariants by explicit Runge–Kutta methods. SIAM J. Sci. Comput. 28(3), 868885 (2006)MathSciNetCrossRefMATH
11.
Calvo, M., Laburta, M.P., Montijano, J.I., Rández, L.: Runge–Kutta. Adv. Comput. Math. 41, 231251 (2015)CrossRefMATH
12.
Cantwell, C.D., Moxey, D., Comerford, A., Bolis, A., Rocco, G., Mengaldo, G., De Grazia, D., Yakovlev, S., Lombard, J.-E., Ekelschot, D., Jordi, B., Xu, H., Mohamied, Y., Eskilsson, C., Nelson, B., Vos, P., Biotto, C., Kirby, R.M., Sherwin, S.J.: Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205–219 (2015)CrossRefMATH
13.
Canuto, C., Hussaini, M., Quarteroni, A., Zang, T.: Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer Verlag, Berlin (2007)MATH
14.
Carey, G.F., Shen, Y.: Approximation of the KdV equation by least square finite elements. J. Comput. Methods Appl. Mech. Eng. 93, 1–11 (1991)CrossRefMATH
15.
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owen, B., Quispel, G.R.W., Wright, W.M.: Energy-preserving Runge–Kutta methods. ESAIM: M2AN 43, 645–649 (2009)MathSciNetCrossRefMATH
16.
Cohen, G., Monk, P.: Gauss point mass lumping schemes for Maxwell’s equations. Numer. Methods Partial Differ. Equ. 14, 63–88 (1998)MathSciNetCrossRefMATH
17.
18.
Cui, Y., Mao, D.K.: Numerical method satisfying the first two conservation laws for the Korteweg-de Vries equation. J. Comput. Phys. 227(1), 376–399 (2007)MathSciNetCrossRefMATH
19.
Deville, M.O., Fischer, P.F., Mund, E.H.: High-order methods for incompressible fluid flow. Cambridge University Press, New York (2002)CrossRefMATH
20.
Dubois-Pelerin, Y., van Kemenade, V., Deville, M.: An object-oriented toolbox for spectral element analysis. J. Sci. Comput. 14(1), 1–29 (1999)MathSciNetCrossRefMATH
21.
Dutykh, D., Katsaounis, T., Mitsotakis, D.: Finite volume methods for unidirectional dispersive wave models. Int. J. Numer. Methods Fluids 71, 717–736 (2013)MathSciNetCrossRef
22.
Dumbser, M., Facchini, M.: A space-time discontinuous Galerkin method for Boussinesq-type equations. Appl. Math. Comput. 272, 336–346 (2016)MathSciNet
23.
24.
Funaro, D.: Spectral Elements for Transport-Dominated Equations. Springer-Verlag, Berlin (1997)CrossRefMATH
25.
Gardner, L.R.T., Gardner, G.A., Ali, A.H.A.: Simulations of solitons using quadratic spline finite elements. J. Comput. Methods Appl. Mech. Eng. 92, 231–243 (1991)MathSciNetCrossRefMATH
26.
Guermond, J.L., Pasquetti, R., Popov, B.: Entropy viscosity method for non-linear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)MathSciNetCrossRefMATH
27.
Holden, H., Karlsen, K.H., Risebro, N.H.: Operator splitting methods for generalized Korteweg-de Vries equations. J. Comput. Phys. 153, 203–222 (1999)MathSciNetCrossRefMATH
28.
Hugues, T.J.R., Feijoo, G.R., Mazzei, L., Quincy, J.B.: The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166, 3–24 (1998)MathSciNetCrossRefMATH
29.
30.
Kappeler, T., Topalov, P.: Global wellposedness of KdV in \(H^{-1} ({\mathbb{T}}, {\mathbb{R}})\). Duke Math. J. 135, 327–360 (2006)MathSciNetCrossRefMATH
31.
Karniadakis, G.E., Sherwin, S.J.: Spectral Hp Element Methods for CFD. Oxford University Press, London (1999)MATH
32.
33.
Kirby, R.M., Karniadakis, G.E.: De-aliasing on non uniform grids: algorithms and applications. J. Comput. Phys. 191, 249–264 (2003)CrossRefMATH
34.
Karpman, V.I.: An asymptotic solution of the Korteweg-de Vries equation. Phys. Lett. 25A(10), 708–709 (1967)CrossRef
35.
Lax, P.D., Levermore, C.D.: The small dispersion limit of the Korteweg-de Vries equation. I. Comm. Pure Appl. Math. XXXVI, 253–290 (1983)MathSciNetCrossRefMATH
36.
Levy, D., Shu, C.W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196, 751–772 (2004)MathSciNetCrossRefMATH
37.
Maday, Y., Patera, A.T.: Spectral element methods for the incompressible Navier–Stokes equations. In: Noor, A.K. (ed.) State-of-the-Art Surveys in Computational Mechanics, pp. 71–143. ASME, New York (1989)
38.
Maday, Y., Kaber, S.M.O., Tadmor, E.: Legendre pseudo-spectral viscosity method for nonlinear conservation laws. SIAM J. Numer. Anal. 30(2), 321–342 (1993)MathSciNetCrossRefMATH
39.
Miles, J.W.: The Korteweg-de Vries equation: a historical essay. J. Fluid Mech. 106, 131–147 (1984)CrossRefMATH
40.
Ohlsson, J.P., Schlatter, P., Fischer, P.F., Henningson, D.S.: Stabilization of the Spectral Element Method in Turbulent Flow Simulations. Lecture Notes in Computational Science and Engineering, vol. 76. Springer-Verlag, Berlin Heidelberg (2011)MATH
41.
Pareschi, L., Russo, G.: Implicit–Explicit Runge–Kutta Schemes for Stiff Systems of Differential Equations, Recent trends in numerical analysis, 269–288. Nova Science Publishers, Inc., Commack (2000)
42.
Samii, A., Panda, N., Michoski, C., Dawson, C.: A hybridized discontinuous Galerkin method for the nonlinear Korteweg-de Vries equation. J. Sci. Comput. 68, 191–212 (2016)MathSciNetCrossRefMATH
43.
44.
45.
Shen, J.: A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KdV equation. SIAM J. Numer. Anal. 41(5), 1595–1619 (2003)MathSciNetCrossRefMATH
46.
47.
Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation. J. Comput. Phys. 55, 231–253 (1984)MathSciNetCrossRefMATH
48.
Vos, P.E.J., Sherwin, S.J., Kirby, R.M.: From \(h\) to \(p\) efficiently: implementing finite and spectral / \(hp\) element methods to achieve optimal performance for low- and high-order discretisations. J. Comput. Phys. 229, 5161–5181 (2010)MathSciNetCrossRefMATH
49.
Walkley, M., Berzins, M.: A finite element method for the one-dimensional extended Boussinesq equations. Int. J. Numer. Meth. Fluids 29, 143–157 (1999)MathSciNetCrossRefMATH
50.
Winther, R.: A conservative finite element method for the Korteweg-de Vries equation. Math. Comput. 34(149), 23–43 (1980)MathSciNetMATH
51.
Xu, C.J., Pasquetti, R.: Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196(2), 680–704 (2004)MathSciNetCrossRefMATH
52.
Xu, Y., Shu, C.W.: Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50(1), 79–104 (2012)MathSciNetCrossRefMATH
53.
Yan, J., Shu, C.-W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17(1–4), 27–47 (2002)MathSciNetCrossRefMATH
54.
55.
Yi, N., Huang, Y., Liu, H.: A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: energy conservation and boundary effect. J. Comput. Phys. 242, 351–366 (2013)MathSciNetCrossRefMATH
56.
Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurence of initial states. Phys. Rev. Lett. 15(6), 240–243 (1965)CrossRefMATH
57.
Zaki, S.I.: A quintic B-spline finite elements scheme for the KdVB equation. J. Comput. Methods Appl. Mech. Eng. 188, 121–134 (2000)CrossRefMATH
58.
Zhao, P., Qin, M.: Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation. J. Phys. A Math. Gen. 33, 3613–3626 (2000)MathSciNetCrossRefMATH
Metadata
Title
High Order -Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and Application to the Korteweg-de Vries Model
Authors
Sebastian Minjeaud
Richard Pasquetti
Publication date
26-05-2017
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-017-0455-2

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