Approximations of the KdV equation by least squares finite element

Abstract The problem of approximating solutions to the Korteweg-de Vries (KdV) equation is investigated using a least squares finite element method. The third order KdV equation is recast as a first-order system and a least-squares finite element approach is introduced for the semidiscrete time-differenced form of the resulting equations. Of particular interest are the approximation properties for solitary wave solutions (solitons). We examine the amplitude and phase error for a representative test problem as well as other examples including the passage of one soliton through another.

[1]  Norman J. Zabusky,et al.  Solitons and Bound States of the Time-Independent Schrödinger Equation , 1968 .

[2]  I. Christie,et al.  A simple adaptive technique for nonlinear wave problems , 1986 .

[3]  R. Hirota.  Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .

[4]  Alan C. Newell,et al.  Solitons in mathematics and physics , 1987 .

[5]  C. Pearson.  Upwind and downwind effects of numerical approximations to wave equations , 1988 .

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

[7]  A. Esq..  XL. Determination of the latitude and longitude of the observatory on the Calton Hill , 1829 .

[8]  Robert M. Miura,et al.  Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation , 1968 .

[9]  Graham F. Carey,et al.  Penalty approximation of stokes flow , 1982 .

[10]  C. Fletcher.  Computational Galerkin Methods , 1983 .

[11]  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 .

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

[13]  Ragnar Winther,et al.  A conservative finite element method for the Korteweg-de Vries equation , 1980 .

[14]  Graham F. Carey,et al.  Least‐squares finite elements for first‐order hyperbolic systems , 1988 .

[15]  Robert L. Lee,et al.  The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2 , 1981 .

[16]  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 .

[17]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[18]  G. Carey,et al.  Least-squares finite element methods for compressible Euler equations , 1990 .