×

A hybridized discontinuous Galerkin method for the nonlinear Korteweg-de Vries equation. (English) Zbl 1344.65093

Summary: In this paper we introduce a hybridized discontinuous Galerkin (HDG) method for solving nonlinear Korteweg-de Vries type equations. Similar to a standard HDG implementation, we first express the approximate variables and numerical fluxes inside each element in terms of the approximate traces of the scalar variable \((u)\), and its first derivative \((u_x)\). These traces are assumed to be single-valued on each face. Next, we impose the conservation of numerical fluxes via two extra sets of equations. Using these global flux conservation conditions and applying the Newton-Raphson method, we construct a system of equations that can be solely expressed in terms of the increments of approximate traces in each iteration. Afterwards, we solve these equations, and substitute the approximate traces back into local equations over each element to obtain local approximate solutions. As for the time stepping scheme, we use the backward difference formulae. The method is proved to be stable for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with \(k\)th order elements, the computed \(u\), \(p\), and \(q\) show optimal convergence at order \(k+1\) in both linear and nonlinear cases, which improves upon previously employed techniques.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexanderian, A., Petra, N., Stadler, G., Ghattas, O.: A fast and scalable method for a-optimal design of experiments for infinite-dimensional bayesian nonlinear inverse problems, p. 29 (2014) · Zbl 06536072
[2] Benzoni-Gavage, S., Danchin, R., Descombes, S., Jamet, D.: Structure of Korteweg models and stability of diffuse interfaces. Interfaces Free Bound. 7(4), 371-414 (2005) · Zbl 1106.35056 · doi:10.4171/IFB/130
[3] Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013) · Zbl 1277.65092
[4] Braginskii, S.I.: Transport processes in a plasma. Rev. Plasma Phys. 1, 205 (1965)
[5] Buckingham, M.: Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments. J. Acoust. Soc. Am. 102(5, 1), 2579-2596 (1997) · doi:10.1121/1.420313
[6] Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42(1), 283-301 (2004) · Zbl 1084.65113 · doi:10.1137/S0036142902417893
[7] Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319-1365 (2009) · Zbl 1205.65312 · doi:10.1137/070706616
[8] Gardner, C.L.: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54(2), 409-427 (1994) · Zbl 0815.35111 · doi:10.1137/S0036139992240425
[9] Heath, R.E., Gamba, I.M., Morrison, P.J., Michler, C.: A discontinuous Galerkin method for the Vlasov-Poisson system. J. Comput. Phys. 231(4), 1140-1174 (2012) · Zbl 1244.82081 · doi:10.1016/j.jcp.2011.09.020
[10] Holmer, J.: The initial-boundary value problem for the Korteweg-de Vries equation. Commun. Partial Differ. Equ. 31(8), 1151-1190 (2006) · Zbl 1111.35062 · doi:10.1080/03605300600718503
[11] Jain, A., Tamma, K.K.: Elliptic heat conduction specialized applications involving high gradients: local discontinuous galerkin finite element method—part 1. J. Therm. Stresses 33(4), 335-343 (2010) · doi:10.1080/01495731003656998
[12] Kaladze, T., Aburjania, G., Kharshiladze, O., Horton, W., Kim, Y.: Theory of magnetized Rossby waves in the ionospheric E layer. J. Geophys. Res. Space Phys · Zbl 1021.65050
[13] Kirby, R.M., Sherwin, S.J., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51(1), 183-212 (2012) · Zbl 1244.65174 · doi:10.1007/s10915-011-9501-7
[14] Kloeckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863-7882 (2009) · Zbl 1175.65111 · doi:10.1016/j.jcp.2009.06.041
[15] Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39, 422-443 (1895) · JFM 26.0881.02 · doi:10.1080/14786449508620739
[16] Kostin, I., Marion, M., Texier-Picard, R., Volpert, V.: Modelling of miscible liquids with the Korteweg stress. ESAIM-Math. Modell. Numer. Anal. 37(5), 741-753 (2003) · Zbl 1201.76045 · doi:10.1051/m2an:2003042
[17] Kubatko, E.J., Westerink, J.J., Dawson, C.: hp discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput. Methods Appl. Mech. Eng. 196(1-3), 437-451 (2006) · Zbl 1120.76348 · doi:10.1016/j.cma.2006.05.002
[18] Larecki, W., Banach, Z.: Influence of nonlinearity of the phonon dispersion relation on wave velocities in the four-moment maximum entropy phonon hydrodynamics. Phys. D-Nonlinear Phenom. 266, 65-79 (2014) · Zbl 1291.82121 · doi:10.1016/j.physd.2013.10.006
[19] Levy, D., Shu, C.-W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196(2), 751-772 (2004) · Zbl 1055.65109 · doi:10.1016/j.jcp.2003.11.013
[20] Loverich, J., Hakim, A., Shumlak, U.: A discontinuous Galerkin method for ideal two-fluid plasma equations. Commun. Comput. Phys. 9(2), 240-268 (2011) · Zbl 1167.76384
[21] Michoski, C., Evans, J.A., Schmitz, P.G., Vasseur, A.: Quantum hydrodynamics with trajectories: the nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry. J. Comput. Phys. 228(23), 8589-8608 (2009) · Zbl 1287.76244 · doi:10.1016/j.jcp.2009.08.011
[22] Michoski, C., Meyerson, D., Isaac, T., Waelbroeck, F.: Discontinuous galerkin methods for plasma physics in the scrape-off layer of tokamaks. J. Comput. Phys. 274, 898-919 (2014) · Zbl 1351.82104 · doi:10.1016/j.jcp.2014.06.058
[23] Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys. 228(23), 8841-8855 (2009) · Zbl 1177.65150 · doi:10.1016/j.jcp.2009.08.030
[24] Panda, N., Dawson, C., Zhang, Y., Kennedy, A.B., Westerink, J.J., Donahue, A.S.: Discontinuous Galerkin methods for solving Boussinesq-Green-Naghdi equations in resolving non-linear and dispersive surface water waves. J. Comput. Phys. 273, 572-588 (2014) · Zbl 1351.76074 · doi:10.1016/j.jcp.2014.05.035
[25] Peraire, J., Nguyen, N., Cockburn, B.: A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations. AIAA Pap. 363, 2010 (2010) · Zbl 1227.76036
[26] Phillips, O.: Nonlinear dispersive waves. Annu. Rev. Fluid Mech. 6, 93-110 (1974) · Zbl 0307.76007 · doi:10.1146/annurev.fl.06.010174.000521
[27] Shukla, P.K., Eliasson, B.: Colloquium: nonlinear collective interactions in quantum plasmas with degenerate electron fluids. Rev. Mod. Phys. 83(3), 885-906 (2011) · doi:10.1103/RevModPhys.83.885
[28] Tassi, E., Morrison, P.J., Waelbroeck, F.L., Grasso, D.: Hamiltonian formulation and analysis of a collisionless fluid reconnection model. Plasma Phys. Control. Fusion 50(8), 085014 (2008)
[29] Woo, S.-B., Choi, Y.-K.: Development of finite volume model for KDV type equation. In: Zuo, Q.H., Dou, X.P., Ge, J.F., (Eds.), Asian and pacific coasts 2007, pp. 203-206, 2007. In: 4th International Conference on Asian and Pacific Coasts, Nanjing Hydraul Res Inst, Nanjing, Peoples Republic of China, Sept 21-24 (2007) · Zbl 1106.35056
[30] Yagi, M., Horton, W.: Reduced Braginskii equations. Phys. Plasmas 1(7), 2135-2139 (1994) · doi:10.1063/1.870611
[31] Yan, J., Shu, C.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769-791 (2002) · Zbl 1021.65050 · doi:10.1137/S0036142901390378
[32] Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal. 19(4), 537-555 (2014) · Zbl 1488.34329 · doi:10.3846/13926292.2014.960016
[33] Zhang, Y., Kennedy, A.B., Panda, N., Dawson, C., Westerink, J.J.: Boussinesq-Green-Naghdi rotational water wave theory. Coast. Eng. 73, 13-27 (2013) · doi:10.1016/j.coastaleng.2012.09.005
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.