115
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Penalty method for the KdV equation

&
Pages 193-211 | Received 20 Feb 2011, Accepted 05 Apr 2011, Published online: 16 Jun 2011

References

  • Ladyženskaya , O . 1952 . On the convergence of Fourier series defining a solution of a mixed problem for hyperbolic equations . Dokl. Akad. Nauk SSSR (N.S.) , 85 : 481 – 484 . (Russian)
  • Rauch , JB and Massey , FJ III . 1974 . Differentiability of solutions to hyperbolic initial-boundary value problems . Trans. Am. Math. Soc. , 189 : 303 – 318 .
  • Smale , S . 1980 . Smooth solutions of heat and wave equations . Comment. Math. Helv. , 55 : 1 – 12 .
  • Temam , R . 1982 . Behaviour at time t = 0 of the solutions of semilinear evolution equations . J. Differ. Eqns , 43 ( 1 ) : 73 – 92 .
  • K.E. Trenberth, Climate System Modeling, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, Press Syndicate of the University of Cambridge, New York, NY, USA, 1992
  • Boyd , JP and Flyer , N . 1999 . Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods (english. english summary) . Methods Appl. Mech. Eng. , 175 ( 3–4 ) : 281 – 309 .
  • Engelman , MS , Sani , RL and Gresho , PM . 1982 . The implementation of normal and/or tangential boundary conditions in finite element codes for incompresible fluid flow . Int. J. Numer. Methods Fluids , 2 ( 3 ) : 225 – 238 . 76–08
  • Gresho , PM . 1991 . Incompressible fluid-dynamics – some fundamental formulation issues . Annual Rev. Fluid Mech. , 23 : 413 – 453 .
  • Gresho , PM and Sani , RL . 1987 . On pressure boundary-conditions for the incompressible navier–stokes equations . Int. J. Numer. Methods Fluids , 7 ( 10 ) : 1111 – 1145 .
  • Heywood , JG . 1980 . Auxiliary flux and pressure conditions for Navier–Stokes problems, in: Approximation mathods for Navier–Stokes problems (proceedings of the symposium in university of Paderborn, Paderborn, 1979) . Lecture Notes Math. , 771 : 223 – 234 .
  • Heywood , JG and Rannacher , R . 1982 . Finite element approximation of the nonstationary Navier–Stokes problem. i. regularity of solutions and second-order error estimates for spatial discretization . SIAM J. Numer. Anal. , 19 ( 2 ) : 275 – 311 .
  • Chen , Q , Qin , Z and Temam , R . Numerical preparation of incompatible initial data for parabolic equation in higher dimension . Math. Comp. , to appear
  • Chen , Q , Qin , Z and Temam , R . 2011 . Numerical resolution near t=0 of nonlinear evolution equations in the presence of corner singularities in space dimension 1 . Commun. Comput. Phys. , 9 ( 3 ) : 568 – 586 .
  • Flyer , N , Qin , Z and Temam , R . Numerical simulations of the penalty method for the KdV and Schrodinger equations , in preparation
  • Flyer , N and Fornberg , B . 2003/04 . On the nature of initial-boundary value solutions for dispersive equations . SIAM J. Appl. Math. , 64 ( 2 ) : 546 – 564 . (electronic)
  • Colin , T and Ghidaglia , J-M . 2001 . An initial-boundary value problem for the Korteweg–de Vries equation posed on a finite interval . Advanced Diff. Eq. , 6 ( 12 ) : 1463 – 1492 .
  • Bona , JL , Sun , SM and Zhang , B-Y . 2001 . The initial-boudary-value problem for the KdV equation on a quater plane . Trans. Am. Math. Soc. , 354 : 427 – 490 .
  • Bona , JL , Sun , SM and Zhang , B-Y . 2003 . A nonhomogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain . Comm. P.D.E. , 28 ( 7,8 ) : 1391 – 1436 .
  • Saut , J-C and Temam , R . 2010 . An initial-boundary value problem for the Zakharov-Kuznetsov equation . Adv. Differ. Eq. , 15 ( 11–12 ) : 1001 – 1031 .
  • Temam , R . 1969 . Sur un problème non-linéaire . J. Math. pures et appl. , 48 : 159 – 172 .
  • Strauss , WA . 1966 . On the continuity of functions with values in various banach space . Pacific J. Math. , 19 ( 3 ) : 543 – 555 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.