Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

Expand
  • 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA;
    2 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, Shandong, China

Received date: 2020-08-30

  Revised date: 2020-11-30

  Online published: 2022-04-29

Supported by

The work of J. Sun and Y. Xing is partially sponsored by NSF grant DMS-1753581.

Abstract

Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, the BBM-BBM system, the Bona-Smith system, etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

Cite this article

Jiawei Sun, Shusen Xie, Yulong Xing . Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System[J]. Communications on Applied Mathematics and Computation, 2022 , 4(2) : 381 -416 . DOI: 10.1007/s42967-021-00119-4

References

1.Amick, C.J.:Regularity and uniqueness of solutions to the Boussinesq system of equations.J.Differ.Equ.54, 231-247 (1984)
2.Antonopoulos, D.C., Dougalis, V.A., Mitsotakis, D.E.:Galerkin approximations of periodic solutions of Boussinesq systems.Bull.Greek Math.Soc.57, 13-30 (2010)
3.Bona, J.L., Chen, M.:A Boussinesq system for two-way propagation of nonlinear dispersive waves.Physica D 116, 191-224 (1998)
4.Bona, J.L., Chen, M.:Singular solutions of a Boussinesq system for water waves.J.Math.Study 49, 205-220 (2016)
5.Bona, J.L., Chen, H., Karakashian, O., Xing, Y.:Conservative, discontinuous-Galerkin methods for the generalized Korteweg-de Vries equation.Math.Comput.82, 1401-1432 (2013)
6.Bona, J.L., Chen, M., Saut, J.-C.:Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media:I.Derivation and linear theory.J.Nonlinear Sci.12, 283-318 (2002)
7.Bona, J.L., Chen, M., Saut, J.-C.:Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media:II.The nonlinear theory.Nonlinearity 17, 925-952 (2004)
8.Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.:Numerical solutions of KdV-KdV systems of Boussinesq equations I.The numerical scheme and generalized solitary waves.Math.Comput.Simul.74, 214-228 (2007)
9.Boussinesq, J.:Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire.Comptes Rendus de l'Acadmie de Sciences 72, 755-759 (1871)
10.Buli, J., Xing, Y.:Local discontinuous Galerkin methods for the Boussinesq coupled BBM system.J.Sci.Comput.75, 536-559 (2018)
11.Burtea, C., Courtès, C.:Discrete energy estimates for the abcd-systems.Commun.Math.Sci.17, 243-298 (2019)
12.Ciarlet, P.:The Finite Element Method for Elliptic Problem.North Holland, USA (1975)
13.Chen, M.:Exact traveling-wave solutions to bidirectional wave equations.Int.J.Theor.Phys.37, 1547-1567 (1998)
14.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)
15.Cockburn, B., Karniadakis, G., Shu, C.-W.:The development of discontinuous Galerkin methods.In:Cockburn B., Karniadakis G.,Shu C.-W., eds.Discontinuous Galerkin Methods:Theory, Computation and Applications.Lecture Notes in Computational Science and Engineering, Part I:Overview, vol.11, pp.3-50 Springer, Berlin, Heidelberg (2000)
16.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)
17.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)
18.Cockburn, B., Shu, C.-W.:The local discontinuous Galerkin finite element method for convectiondiffusion systems.SIAM J.Numer.Anal.35, 2440-2463 (1998)
19.Gottlieb, S., Shu, C.-W., Tadmor, E.:Strong stability-preserving high-order time discretization methods.SIAM Rev.43, 89-112 (2001)
20.Hufford, C., Xing, Y.:Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation.J.Comput.Appl.Math.255, 441-455 (2014)
21.Karakashian, O., Xing, Y.:A posteriori error estimates for conservative local discontinuous Galerkin methods for the generalized Korteweg-de Vries equation.Commun.Comput.Phys.20, 250-278 (2016)
22.Li, X., Xing, Y., Chou, C.-S.:Optimal energy conserving and energy dissipative local discontinuous Galerkin methods for the Benjamin-Bona-Mahony equation.J.Sci.Comput.83, 17 (2020)
23.Li, X., Sun, W., Xing, Y., Chou, C.-S.:Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation.Journal of Computational Physics 401, 109002 (2020)
24.Luo, J., Shu, C.-W., Zhang, Q.:A priori error estimates to smooth solutions of the third order RungeKutta discontinuous Galerkin method for symmetrizable systems of conservation laws.ESAIM:M2AN 49, 991-1018 (2015)
25.Peregrine, D.H.:Calculations of the development of an undular bore.J.Fluid Mech.25, 321-330 (1966)
26.Xu, Y., Shu, C.-W.:Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations.Comput.Methods Appl.Mech.Eng.196, 3805-3822 (2007)
27.Xu, Y., Shu, C.-W.:Local discontinuous Galerkin methods for high-order time-dependent partial differential equations.Commun.Comput.Phys.7, 1-46 (2010)
28.Xu, Y., 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)
Options
Outlines

/