Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

Abstract

In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

Keywords

Local discontinuous Galerkin method / Two-dimensional Camassa–Holm equation / Stability

References

[1] Artebrant R, Schroll HJ. Numerical simulation of Camassa Holm peakons by adaptive upwinding. Appl. Numer. Math.. 2006, 56 695-711

[2] Bona JL, Chen H, Karakashian O, Xing Y. Conservative, discontinuous-Galerkin methods for the generalized Korteweg–de Vries equation. Math. Comput.. 2013, 82 1401-1432

[3] Cai W, Sun Y, Wang Y. Geometric numerical integration for peakon b-family equations. Commun. Comput. Phys.. 2016, 19 24-52

[4] Camassa R, Kuang D, Lee L. Solitary waves and N-particle algorithms for a class of Euler–Poincaré equations. Stud. Appl. Math.. 2016, 137 502-546

[5] Cao HY, Sun ZZ, Gao GH. A three-level linearized finite difference scheme for the Camassa–Holm equation. Numer. Methods Partial Differ. Eq.. 2014, 30 451-471

[6] Chertock A, Du Toit P, Marsden JE. Integration of the EPDiff equation by particle methods, ESAIM. Math. Model. Numer. Anal.. 2012, 46 515-534

[7] Chertock A, Liu JG, Pendleton T. Convergence of a particle method and global weak solutions of a family of evolutionary PDEs. SIAM J. Numer. Anal.. 2012, 50 1-21

[8] Chiu PH, Lee L, Sheu TWH. A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation. J. Comput. Phys.. 2009, 228 8034-8052

[9] Chiu PH, Lee L, Sheu TWH. A sixth-order dual preserving algorithm for the Camassa–Holm equation. J. Comput. Appl. Math.. 2010, 233 2767-2778

[10] Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods, in n Discontinuous Galerkin Methods: Theory, Computation and Applications, Cockburn, B., Karniadakis, G., Shu, C.-W., editors, Lecture Notes in Computational Science and Engineering, vol. 11, Springer, Berlin, Part I: Overview, pp. 3–50 (2000)

[11] Cockburn B, Shu C-W. Runge–Kutta Discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput.. 2001, 16 173-261

[12] Cockburn B, Shu C-W. Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput.. 2005, 22 23 1-3

[13] Coclite GM, Karlsen KH, Risebro NH. An explicit finite difference scheme for the Camassa–Holm equation. Adv. Differ. Eq.. 2008, 13 681-732

[14] Cotter, C., Holm, D.: Momentum Maps for Lattice EPDiff. Handbook of Numerical Analysis. Vol. XIV. Special volume: Computational Methods for the Atmosphere and the Oceans, pp. 247–278, Handb. Numer. Anal., 14, Elsevier/North-Holland, Amsterdam, (2009)

[15] Feng BF, Maruno K, Ohta Y. A self-adaptive moving mesh method for the Camassa–Holm equation. J. Comput. Appl. Math.. 2010, 235 229-243

[16] Gong Y, Wang Y. An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs. Commun. Comput. Phys.. 2016, 20 1313-1339

[17] Holm D, Marsden J. Momentum Maps and Measure-Valued Solutions (Peakons, Filaments, and Sheets) for the EPDiff Equation, The Breadth of Symplectic and Poisson Geometry, 203–235, Progress in Mathematics, 232. 2005 Boston: Birkhäuser Boston

[18] Holm, D., Schmah, T., Stoica, C.: Geometric mechanics and symmetry. From finite to infinite dimensions. With solutions to selected exercises by David Ellis, C.P., Oxford Texts in Applied and Engineering Mathematics, 12. Oxford University Press, Oxford, (2009)

[19] Holden H, Raynaud X. A convergent numerical scheme for the Camassa–Holm equation based on multipeakons. Discrete Contin. Dyn. Syst.. 2006, 14 505-523

[20] Holden H, Raynaud X. Convergence of a finite difference scheme for the Camassa–Holm equation. SIAM J. Numer. Anal.. 2006, 44 1655-1680

[21] Kraenkel RA, Zenchuk AI. Two-dimensional integrable generalization of the Camassa–Holm equation. Phys. Lett. A. 1999, 260 218-224

[22] Kraenkel RA, Senthilvelan M, Zenchuk AI. Lie symmetry analysis and reductions of a two-dimensional integrable generalization of the Camassa–Holm equation. Phys. Lett. A. 2000, 273 183-193

[23] Kruse H-P, Scheurle J, Du W. A Two-Dimensional Version of the Camassa–Holm Equation, Symmetry and Perturbation Theory. 2001 River Edge: World Science Publisher. 120-127

[24] Kalisch H, Lenells J. Numerical study of traveling-wave solutions for the Camassa–Holm equation. Chaos Solitons Fractals. 2005, 25 287-298

[25] Kalisch H, Raynaud X. Convergence of a spectral projection of the Camassa–Holm equation. Numer. Methods Partial Differ. Eq.. 2006, 22 1197-1215

[26] Li M, Chen A. High order central discontinuous Galerkin-finite element methods for the Camassa–Holm equation. Appl. Math. Comput.. 2014, 227 237-245

[27] Liu H, Pendleton T. On invariant-preserving finite difference schemes for the Camassa–Holm equation and the two-component Camassa–Holm system. Commun. Comput. Phys.. 2016, 19 1015-1041

[28] Liu H, Xing Y. An invariant preserving discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Sci. Comput.. 2016, 38 A1919-A1934

[29] Matsuo T. A Hamiltonian-conserving Galerkin scheme for the Camassa–Holm equation. J. Comput. Appl. Math.. 2010, 234 1258-1266

[30] Miyatake Y, Matsuo T. Energy-preserving $H^1$-Galerkin schemes for shallow water wave equations with peakon solutions. Phys. Lett. A. 2012, 376 2633-2639

[31] Shu C-W, Osher S. Efficient implementation of essentially nonoscillatory shock capturing schemes. J. Comput. Phys.. 1988, 77 439-471

[32] Wang ZQ, Xiang XX. Generalized Laguerre approximations and spectral method for the Camassa–Holm equation. IMA J. Numer. Anal.. 2015, 35 1456-1482

[33] Xu Y, Shu C-W. A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Numer. Anal.. 2008, 46 1998-2021

[34] Xu Y, Shu C-W. Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys.. 2010, 7 1-46

[35] Yu CH, Sheu TWH, Chang CH, Liao SJ. Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa–Holm equation with different initial solitary waves. Numer. Methods Partial Differ. Equ.. 2015, 31 1645-1664

[36] Yu CH, Sheu TWH. Development of a combined compact difference scheme to simulate soliton collision in a shallow water equation. Commun. Comput. Phys.. 2016, 19 603-631

[37] Yu CH, Sheu TWH. Numerical study of long-time Camassa–Holm solution behavior for soliton transport. Math. Comput. Simul.. 2016, 128 1-12

[38] Zhang Y, Deng ZC, Hu WP. Multisymplectic method for the Camassa-Holm equation. Adv. Differ. Eq.. 2016, 2016 7

Funding

National Natural Science Foundation of China(11722112)