Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations

Chen, Yanlai; Dong, Bo; Jiang, Jiahua

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2283-2306.

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.

[1] T.R. Akylas and T.-S. Yang, On short-scale oscillatory tails of long-wave disturbances. Stud. Appl. Math. 94 (1995) 1–20. | DOI | MR | Zbl

[2] J.T. Beale, Exact solitary waves with capillary ripples at infinity. Commun. Pure Appl. Math. 44 (1991) 211–247. | DOI | MR | Zbl

[3] J.P. Boyd, Weak non-local solitons for capillary-gravity waves: fifth-order Korteweg-de Vries equation. Physica D 48 (1991) 129–146. | DOI | Zbl

[4] J.L. Bona, H. Chen, O. Karakashian and Y. Xing, Conservative, discontinuous Galerkin-methods for the generalized Korteweg de Vries equation. Math. Comput. 82 (2013) 1401–1432. | DOI | MR | Zbl

[5] Y. Chen, B. Cockburn and B. Dong, A new discontinuous Galerkin method, conserving the discrete H²-norm, for third-order linear equations in one space dimension. IMA J. Numer. Anal. 36 (2016) 1570–1598. | DOI | MR | Zbl

[6] Y. Chen, B. Cockburn and B. Dong, Superconvergent HDG methods for linear, stationary, third-order equations in one-space dimension. Math. Comput. 85 (2016) 2715–2742. | DOI | MR | Zbl

[7] Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77 (2008) 699–730. | DOI | MR | Zbl

[8] B. Cockburn, J. Gopalakrishnan and F.-J. Sayas, A projection-based error analysis of HDG methods. Math. Comput. 79 (2010) 1351–1367. | DOI | MR | Zbl

[9] B. Dong, Optimally convergent HDG method for third-order Korteweg-de Vries type equations. J. Sci. Comput. 73 (2017) 712–735. | DOI | MR | Zbl

[10] R. Guo and Y. Xu, Fast solver for the local discontinuous Galerkin discretization of the KdV type equations. Commun. Comput. Phys. 17 (2015) 424–457. | DOI | MR | Zbl

[11] C. Hufford and Y. Xing, Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg-de Vries equation. J. Comput. Appl. Math. 255 (2014) 441–455. | DOI | MR | Zbl

[12] J.K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D 32 (1988) 253–268. | DOI | MR | Zbl

[13] T. Kakutani and H. Ono, Weak nonlinear hydromagnetic waves in a cold collision-free plasma. J. Phys. Soc. Jpn. 26 (1969) 1305–1318. | DOI

[14] O. Karakashian and Y. Xing, A posteriori error estimates for conservative local discontinuous Galerkin methods for the Generalized Korteweg-de Vries equation. Commun. Comput. Phys. 20 (2016) 250–278. | DOI | MR | Zbl

[15] T. Kato, Low regularity well-posedness for the periodic Kawahara equation. Differ. Integral Equ. 25 (2012) 1011–1036. | MR | Zbl

[16] T. Kawahara, Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33 (1972) 260–264. | DOI

[17] H. Nagashima, Experiment on solitary waves in the nonlinear transmission line described by the equation $\frac{\theta u}{\theta \tau }+u\frac{\theta u}{\theta \xi }-\frac{{\theta }^{5}u}{\theta {\xi }^5}=0$. J. Phys. Soc. Jpn. 47 (1979) 1387–1388. | DOI

[18] N.C. Nguyen, J. Peraire and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys. 228 (2009) 8841–8855. | DOI | MR | Zbl

[19] Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for three classes of nonlinear wave equations. Special issue dedicated to the 70th birthday of Professor Zhong-Ci Shi. J. Comput. Math. 22 (2004) 250–274. | MR | Zbl

[20] Y. Xu and C.-W. Shu, Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3805–3822. | DOI | MR | Zbl

[21] Y. Xu and C.-W. Shu, Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50 (2012) 79–104. | DOI | MR | Zbl

[22] J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769–791. | DOI | MR | Zbl