A first-order hyperbolic system approach for dispersion

We propose a new first-order hyperbolic system approach for dispersive partial differential equations.We apply a compact 4th-order RD scheme, and solve time dependent dispersive PDEs.We demonstrate the performance of the high-order RD schemes on the proposed hyperbolic system, including dispersive shock cases.We verify that the predicted solution, its gradient and Hessian have the same order of accuracy on randomly distributed nodes.

[1]  Sylfest Glimsdal,et al.  Dispersion of tsunamis: does it really matter? , 2013 .

[2]  Eleuterio F. Toro,et al.  Advection-Diffusion-Reaction Equations: Hyperbolization and High-Order ADER Discretizations , 2014, SIAM J. Sci. Comput..

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

[4]  A. Debussche,et al.  Numerical simulation of the stochastic Korteweg-de Vries equation , 1999 .

[5]  Greg Roff,et al.  The Morning Glory: An extraordinary atmospheric undular bore , 1982 .

[6]  Alireza Mazaheri,et al.  Very efficient high-order hyperbolic schemes for time-dependent advection–diffusion problems: Third-, fourth-, and sixth-order , 2014 .

[7]  Hiroaki Nishikawa,et al.  Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids , 2015, J. Comput. Phys..

[8]  Kung-Chien Wu,et al.  Hydrodynamic limits of the nonlinear Klein–Gordon equation , 2012 .

[9]  Mario Ricchiuto,et al.  On the nonlinear behaviour of Boussinesq type models: Amplitude-velocity vs amplitude-flux forms , 2015 .

[10]  Henrik Kalisch,et al.  A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods , 2009, Math. Comput. Simul..

[11]  Sylvie Benzoni-Gavage,et al.  Planar traveling waves in capillary fluids , 2013, Differential and Integral Equations.

[12]  I. Coddington,et al.  Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics , 2006 .

[13]  M. A. Hoefer,et al.  Shock Waves in Dispersive Eulerian Fluids , 2013, J. Nonlinear Sci..

[14]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. II: Unification of advection and diffusion , 2010, J. Comput. Phys..

[15]  Jean-Claude Saut,et al.  Madelung, Gross–Pitaevskii and Korteweg , 2011, 1111.4670.

[16]  Jason W. Fleischer,et al.  Dispersive, superfluid-like shock waves in nonlinear optics: Properties & interactions , 2007, 2007 Quantum Electronics and Laser Science Conference.

[17]  Hailiang Liu,et al.  A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation: Energy conservation and boundary effect , 2013, J. Comput. Phys..

[18]  Hubert Chanson,et al.  Current knowledge in hydraulic jumps and related phenomena: A survey of experimental results , 2009 .

[19]  Roger H.J. Grimshaw,et al.  Atmospheric Internal Solitary Waves , 2003 .

[20]  Alireza Mazaheri,et al.  Improved second-order hyperbolic residual-distribution scheme and its extension to third-order on arbitrary triangular grids , 2015, J. Comput. Phys..

[21]  Nicholas K. Lowman,et al.  Dispersive shock waves in viscously deformable media , 2013, Journal of Fluid Mechanics.

[22]  Hervé Leblond,et al.  Interaction of two solitary waves in a ferromagnet , 1995 .

[23]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes , 2007, J. Comput. Phys..

[24]  David Lannes,et al.  The Water Waves Problem: Mathematical Analysis and Asymptotics , 2013 .

[25]  Hailiang Liu,et al.  A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect , 2006, J. Comput. Phys..

[26]  Ying Wang,et al.  A Fast Explicit Operator Splitting Method for Modified Buckley–Leverett Equations , 2015, J. Sci. Comput..

[27]  Alireza Mazaheri,et al.  First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems , 2014 .

[28]  Giancarlo Ruocco,et al.  Shocks in nonlocal media. , 2007, Physical review letters.