A boundary value problem for the KdV equation: Comparison of finite-difference and Chebyshev methods

Solutions of a boundary value problem for the Korteweg-de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient.

[1]  Bing-Yu Zhang,et al.  A Nonhomogeneous Boundary-Value Problem for the Korteweg–de Vries Equation Posed on a Finite Domain , 2003 .

[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]  Bing-Yu Zhang,et al.  A non-homogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain II , 2003 .

[4]  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.

[5]  D. Crighton.  Applications of KdV , 1995 .

[6]  D. G. Cacuci,et al.  Benchmarking the propagator method for nonlinear systems: a Burgers-Korteweg-Devries equation , 1990 .

[7]  J. E. Colliander,et al.  THE GENERALIZED KORTEWEG–DE VRIES EQUATION ON THE HALF LINE , 2001 .

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

[9]  D. Korteweg,et al.  On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 2011 .

[10]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[11]  Wilhelm Heinrichs,et al.  Spectral Approximation of Third-Order Problems , 1999, J. Sci. Comput..

[12]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[13]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[14]  A. Jeffrey,et al.  Weak Nonlinear Dispersive Waves: A Discussion Centered Around the Korteweg–De Vries Equation , 1972 .

[15]  T. Driscoll,et al.  Regular Article: A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion , 1999 .

[16]  Xavier Raynaud,et al.  ON THE RATE OF CONVERGENCE OF A COLLOCATION PROJECTION OF THE KDV EQUATION , 2007 .

[17]  Thiab R. Taha,et al.  Analytical and numerical aspects of certain nonlinear evolution equations. 1V. numerical modified Korteweg-de Vries equation , 1988 .

[18]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[19]  A. Quarteroni,et al.  Error analysis for spectral approximation of the Korteweg-De Vries equation , 1988 .

[20]  Jie Shen,et al.  A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation , 2003, SIAM J. Numer. Anal..

[21]  Graham F. Carey,et al.  Approximations of the KdV equation by least squares finite element , 1991 .

[22]  Jean-Michel Ghidaglia,et al.  An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval , 2001, Advances in Differential Equations.

[23]  J. M. Sanz-Serna,et al.  An easily implementable fourth-order method for the time integration of wave problems , 1992 .

[24]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[25]  Weiwei Sun,et al.  A Legendre-Petrov-Galerkin and Chebyshev Collocation Method for Third-Order Differential Equations , 2000, SIAM J. Numer. Anal..

[26]  Edward H. Twizell,et al.  Numerical methods for the solution of the third- and fifth-order dispersive Korteweg-de Vries equations , 1995 .

[27]  D. Pavoni,et al.  Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation , 1988 .

[28]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[29]  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.

[30]  P. Drazin,et al.  Solitons: An Introduction , 1989 .