A Hybridized Discontinuous Galerkin Method for the Nonlinear Korteweg–de Vries Equation

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 ($$u_x$$ux). 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 kth order elements, the computed u, p, and q show optimal convergence at order $$k+1$$k+1 in both linear and nonlinear cases, which improves upon previously employed techniques.

[1]  P. J. Morrison,et al.  A discontinuous Galerkin method for the Vlasov-Poisson system , 2010, J. Comput. Phys..

[2]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[3]  Carl L. Gardner,et al.  The Quantum Hydrodynamic Model for Semiconductor Devices , 1994, SIAM J. Appl. Math..

[4]  Doron Levy,et al.  Local discontinuous Galerkin methods for nonlinear dispersive equations , 2004 .

[5]  O M Phillips,et al.  Nonlinear Dispersive Waves , 1974 .

[6]  Craig Michoski,et al.  Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry , 2009, J. Comput. Phys..

[7]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[8]  Ethan J. Kubatko,et al.  hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow , 2006 .

[9]  Kumar K. Tamma,et al.  Elliptic Heat Conduction Specialized Applications Involving High Gradients: Local Discontinuous Galerkin Finite Element Method—Part 1 , 2010 .

[10]  P. Shukla,et al.  Colloquium : nonlinear collective interactions in quantum plasmas with degenerate electron fluids , 2010, 1009.5215.

[11]  Yao Zhang,et al.  Discontinuous Galerkin methods for solving Boussinesq-Green-Naghdi equations in resolving non-linear and dispersive surface water waves , 2014, J. Comput. Phys..

[12]  Robert Michael Kirby,et al.  To CG or to HDG: A Comparative Study , 2012, J. Sci. Comput..

[13]  Bernardo Cockburn,et al.  A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations , 2010, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition.

[14]  Bernardo Cockburn,et al.  A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems , 2004, SIAM J. Numer. Anal..

[15]  Chi-Wang Shu,et al.  A Local Discontinuous Galerkin Method for KdV Type Equations , 2002, SIAM J. Numer. Anal..

[16]  Bernardo Cockburn,et al.  A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations , 2010 .

[17]  W. Horton,et al.  Theory of magnetized Rossby waves in the ionospheric E layer , 2004 .

[18]  Uri Shumlak,et al.  A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations , 2010, 1003.4542.

[19]  Justin Holmer.  The Initial-Boundary Value Problem for the Korteweg–de Vries Equation , 2005 .

[20]  W. Horton,et al.  REDUCED BRAGINSKII EQUATIONS , 1993 .

[21]  P. Morrison,et al.  Hamiltonian formulation and analysis of a collisionless fluid reconnection model , 2008, 0802.3849.

[22]  Bernardo Cockburn,et al.  An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations , 2009, J. Comput. Phys..

[23]  S. I. Braginskii.  Transport Processes in a Plasma , 1965 .

[24]  Z. Banach,et al.  Influence of nonlinearity of the phonon dispersion relation on wave velocities in the four-moment maximum entropy phonon hydrodynamics , 2014 .

[25]  M. Buckingham.  Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments , 1997 .

[26]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[27]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[28]  Ferdinando Auricchio,et al.  Mixed Finite Element Methods , 2004 .

[29]  S. Descombes,et al.  Structure of Korteweg models and stability of diffuse interfaces , 2005 .

[30]  Zheng-rong Liu,et al.  Existence of Solitary Waves and Periodic Waves to a Perturbed Generalized KdV Equation , 2014 .

[31]  Craig Michoski,et al.  Discontinuous Galerkin methods for plasma physics in the scrape-off layer of tokamaks , 2014, J. Comput. Phys..

[32]  Joannes J. Westerink,et al.  Boussinesq-Green-Naghdi rotational water wave theory , 2013 .

[33]  M. Marion,et al.  Modelling of Miscible Liquids with the Korteweg Stress , 2003 .

[34]  Georg Stadler,et al.  A Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-dimensional Bayesian Nonlinear Inverse Problems , 2014, SIAM J. Sci. Comput..