×

zbMATH — the first resource for mathematics

Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. (English) Zbl 1124.76035
Summary: We develop a local discontinuous Galerkin method to solve Kuramoto-Sivashinsky equations and ItĂ´-type coupled KdV equations. The \(L^{2}\) stability of the schemes is obtained for both of these nonlinear equations. We use both the traditional nonlinearly stable explicit high-order Runge-Kutta methods and the explicit exponential time differencing method for the time discretization; the latter can achieve high-order accuracy and maintain good stability while avoiding the very restrictive explicit stability limit of the former when the PDE contains higher-order spatial derivatives. Numerical examples demonstrate accuracy and capability of these methods.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Antonowicz, M.; Fordy, A.P., Coupled KdV equations with multi-Hamiltonian structures, Physica D, 28, 345-357, (1987) · Zbl 0638.35079
[2] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. comput. phys., 131, 267-279, (1997) · Zbl 0871.76040
[3] Beylkin, G.; Keiser, J.M.; Vozovoi, L., A new class of time discretization schemes for the solution of nonlinear pdes, J. comput. phys., 147, 362-387, (1998) · Zbl 0924.65089
[4] Cockburn, B., Discontinuous Galerkin methods for methods for convection-dominated problems, (), 69-224 · Zbl 0937.76049
[5] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comput., 54, 545-581, (1990) · Zbl 0695.65066
[6] Cockburn, B.; Karniadakis, G.; Shu, C.-W., The development of discontinuous Galerkin methods, (), 3-50, Part I: Overview · Zbl 0989.76045
[7] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093
[8] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. comput., 52, 411-435, (1989) · Zbl 0662.65083
[9] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. comput. phys., 141, 199-224, (1998) · Zbl 0920.65059
[10] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. numer. anal., 35, 2440-2463, (1998) · Zbl 0927.65118
[11] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[12] Cox, S.M.; Matthews, P.C., Exponential time differencing for stiff systems, J. comput. phys., 176, 430-455, (2002) · Zbl 1005.65069
[13] Hooper, A.P.; Grimshaw, R., Travelling wave solutions of the Kuramoto-Sivashinsky equation, Wave motion, 10, 405-420, (1988) · Zbl 0692.35142
[14] Hyman, J.M.; Nicolaenko, B., The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems, Physica D, 18, 113-126, (1986) · Zbl 0602.58033
[15] Ito, M., Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. lett. A, 91, 335-338, (1982)
[16] Kassam, A.K.; Trefethen, L.N., Fourth-order time-stepping for stiff pdes, SIAM J. sci. comput., 26, 1214-1233, (2005) · Zbl 1077.65105
[17] Kupershmidt, B.A., A coupled Korteweg-de Vries equation with dispersion, J. phys. A: math. gen., 18, L571-L573, (1985) · Zbl 0586.35082
[18] Levy, D.; Shu, C.-W.; Yan, J., Local discontinuous Galerkin methods for nonlinear dispersive equations, J. comput. phys., 196, 751-772, (2004) · Zbl 1055.65109
[19] Michelson, D., Steady solutions of the Kuramoto-Sivashinsky equation, Physica D, 19, 89-111, (1986) · Zbl 0603.35080
[20] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[21] Tadmor, E., The well-posedness of the Kuramoto-Sivashinsky equation, SIAM J. math. anal., 17, 884-893, (1986) · Zbl 0606.35073
[22] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001
[23] Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for three classes of nonlinear wave equations, J. comput. math., 22, 250-274, (2004) · Zbl 1050.65093
[24] Yan, J.; Shu, C.-W., A local discontinuous Galerkin method for KdV type equations, SIAM J. numer. anal., 40, 769-791, (2002) · Zbl 1021.65050
[25] Yan, J.; Shu, C.-W., Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. sci. comput., 17, 27-47, (2002) · Zbl 1003.65115
[26] Yang, T.S., On traveling-wave solutions of the Kuramoto-Sivashinsky equation, Physica D, 110, 25-42, (1997) · Zbl 0925.35136
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.