• Received : 2011.09.13
  • Accepted : 2011.12.13
  • Published : 2011.12.25


We consider the second order Strang splitting method to approximate the solution to the KdV equation. The model equation is split into three sets of initial value problems containing convection and dispersal terms separately. TVD MUSCL or MUSCL scheme is applied to approximate the convection term and the second order centered difference method to approximate the dispersal term. In time stepping, explicit third order Runge-Kutta method is used to the equation containing convection term and implicit Crank-Nicolson method to the equation containing dispersal term to reduce the CFL restriction. Several numerical examples of weakly and strongly dispersive problems, which produce solitons or dispersive shock waves, or may show instabilities of the solution, are presented.


Supported by : NRF


  1. U. M. Ascher and R. I. McLachlan, On symplectic and Multisymplectic schemes for the KdVequation, Journal of Scientific Computing 25, (2005), 83-104.
  2. K. Djidjeli, W. G. Price, E. H. Twizell and Y. Wang, Numerical methods for the solution of the third- and fifth-order dispersive Korteweg-de Vries equations, Journal of Computational and Applied Mathematics 58, (1995), 307-336.
  3. S. Gottlieb and C. -W. Shu, Total variation diminishing Runge-Kutta schemes, Mathematics of Computation 67, (1998), no. 221, 73-85.
  4. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels and V. Sehweikhard, Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics, Physical Review A 74,(2006), 023623- 1-023623-24.
  5. C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, 1987.
  6. R. J. Leveque, Finte volume methods for hyperbolic prohlems, Cambridge University Press, 2002.
  7. S. Osher, Riemann solvers, the entropy condition, and dijfference approximations, SIAM J. Numer. Anal. 21, (1984), no. 21, 217-235.
  8. S. Osher, Convergence of generalized MUSCL, schemes, SIAM J. Numer. Anal. 22, (1985), no. 5, 217-235.
  9. J. C. Tannchill, D. A. Anderson and R. H. PIetcher, Computational fluid mechanics and heat transfer Taylor & Francis, 1997.
  10. E. F. Toro, Riemann solvers and numerical methods for fluid dynamics, Springer, 1999.
  11. J. Yan and C. -W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM Journal on Numerical Analysis 40, (2002), no. 2, 769-791.