Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations

In this paper, we develop, analyze and test local discontinuous Galerkin (DG) methods to solve two classes of two-dimensional nonlinear wave equations formulated by the Kadomtsev–Petviashvili (KP) equation and the Zakharov–Kuznetsov (ZK) equation. Our proposed scheme for the Kadomtsev–Petviashvili equation satisfies the constraint from the PDE which contains a non-local operator and at the same time has the local property of the discontinuous Galerkin methods. The scheme for the Zakharov–Kuznetsov equation extends the previous work on local discontinuous Galerkin method solving one-dimensional nonlinear wave equations to the two-dimensional case. L2 stability of the schemes is proved for both of these two nonlinear equations. Numerical examples are shown to illustrate the accuracy and capability of the methods.

[1]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[2]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[3]  Toh,et al.  Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion. , 1989, Physical review. A, General physics.

[4]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  Takuji Kawahara,et al.  Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation , 1990 .

[6]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives , 2002, J. Sci. Comput..

[7]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water , 1989, Journal of Fluid Mechanics.

[8]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[9]  Boris Dubrovin,et al.  Three‐Phase Solutions of the Kadomtsev–Petviashvili Equation , 1997 .

[10]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[11]  Norman W. Scheffner,et al.  Two-dimensional periodic waves in shallow water. Part 2. Asymmetric waves , 1995, Journal of Fluid Mechanics.

[12]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[13]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[14]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[15]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[16]  K. Nozaki.  Vortex Solitons of Drift Waves and Anomalous Diffusion , 1981 .

[17]  Spencer J. Sherwin,et al.  A Discontinuous Spectral Element Model for Boussinesq-Type Equations , 2002, J. Sci. Comput..

[18]  Solitary internal waves in a rotating channel: A numerical study , 1987 .

[19]  YanXu,et al.  LOCAL DISCONTINUOUS GALERKIN METHODS FOR THREE CLASSES OF NONLINEAR WAVE EQUATIONS , 2004 .

[20]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[21]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[22]  T. R. Akylas,et al.  On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects , 1987, Journal of Fluid Mechanics.

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

[24]  Bernard Deconinck,et al.  The KP equation with quasiperiodic initial data , 1998 .

[25]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[26]  T. Mitsui,et al.  A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations , 1999 .

[27]  Jean-Claude Saut,et al.  Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation , 2002 .

[28]  H. Takaoka,et al.  On the local regularity of the Kadomtsev-Petviashvili-II equation , 2001 .

[29]  H. Segur,et al.  An Analytical Model of Periodic Waves in Shallow Water , 1985 .

[30]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[31]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[32]  Vladimir E. Zakharov,et al.  Three-dimensional solitons , 1974 .

[33]  Some Generalizations of the Kadomtsev–Petviashvili Equations☆☆☆ , 2000 .

[34]  F. Smith,et al.  Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[35]  L. Ridgway Scott,et al.  Implicit spectral methods for wave propagation problems , 1991 .

[36]  Andrei V. Faminskii,et al.  The Cauchy problem for the Kadomtsev-Petviashvili equation , 1990 .

[37]  V. Petviashvili,et al.  Red spot of Jupiter and the drift soliton in a plasma , 1980 .

[38]  Mark J. Ablowitz,et al.  Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation , 1994 .

[39]  Bao-Feng Feng,et al.  A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations , 1998 .

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

[41]  Jean Bourgain,et al.  On the Cauchy problem for the Kadomstev-Petviashvili equation , 1993 .

[42]  John Argyris,et al.  An engineer's guide to soliton phenomena: Application of the finite element method , 1987 .

[43]  P. Isaza,et al.  Local Solution for the Kadomtsev-Petviashvili Equation in R2 , 1995 .

[44]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..