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Operator splitting for partial differential equations with Burgers nonlinearity

Authors: Helge Holden, Christian Lubich and Nils Henrik Risebro
Journal: Math. Comp. 82 (2013), 173-185
MSC (2010): Primary 35Q53; Secondary 65M12, 65M15
Published electronically: June 12, 2012
MathSciNet review: 2983020
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Abstract | References | Similar Articles | Additional Information

Abstract: We provide a new analytical approach to operator splitting for equations of the type $ u_t=Au+u u_x$ where $ A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $ H^r$ for initial data in $ H^{r+5}$ with arbitrary $ r\ge 1$.

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  • 1. A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis. Cambridge UP, Cambridge, 1995.
  • 2. D. J. Benney. Long waves on liquid films. J. Math. and Phys. 45:150-155 (1966). MR 0201125 (34:1010)
  • 3. H. A. Biagioni and F. Linares. On the Benney-Lin and Kawahara equations. J. Math. Anal. Appl. 211:131-152 (1997). MR 1460163 (98e:35140)
  • 4. J. L. Bona and R. Smith. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 278:555-601 (1975). MR 0385355 (52:6219)
  • 5. E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems. Second edition. Springer, Berlin, 1993. MR 1227985 (94c:65005)
  • 6. H. Holden, K. H. Karlsen, K.-A. Lie, and N. H. Risebro. Splitting for Partial Differential Equations with Rough Solutions. European Math. Soc. Publishing House, Zürich, 2010. MR 2662342 (2011j:65002)
  • 7. H. Holden, K. H. Karlsen, and N. H. Risebro. Operator splitting methods for generalized Korteweg-de Vries equations. J. Comput. Phys. 153:203-222 (1999). MR 1703652 (2001c:65101)
  • 8. H. Holden, K. H. Karlsen, N. H. Risebro, and T. Tao. Operator splitting methods for the Korteweg-de Vries equation. Math. Comp. 80:821-846 (2011). MR 2772097
  • 9. H. Holden, K. H. Karlsen, T. Karper. Operator splitting for two-dimensional incompressible fluid equations. Math. Comp., to appear.
  • 10. T. Jahnke and C. Lubich. Error bounds for exponential operator splittings. BIT 40:735-744 (2000). MR 1799313 (2001k:65143)
  • 11. Y. Jia and Z. Huo. Well-posedness for the fifth-order shallow water equations. J. Diff. Eqn. 246:2448-2467 (2009). MR 2498848 (2010k:35411)
  • 12. T. Kawahara. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33:260-264 (1975).
  • 13. C. E. Kenig, G. Ponce, and L. Vega. Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. 4:323-347 (1991). MR 1086966 (92c:35106)
  • 14. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva. Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, 1968. MR 0241821 (39:3159a)
  • 15. S. P. Lin. Finite amplitude side-band stability of a viscous film. J. Fluid. Mech. 63:417-429 (1974).
  • 16. F. Linares and G. Ponce. Introduction to Nonlinear Dispersive Equations. Springer, 2009. MR 2492151 (2010j:35001)
  • 17. C. Lubich. On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77:2141-2153 (2008). MR 2429878 (2009d:65114)
  • 18. T. Tao. Nonlinear Dispersive Equations. Local and Global Analysis. Amer. Math. Soc., Providence, 2006. MR 2233925 (2008i:35211)
  • 19. F. Tappert. Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-step Fourier method. In: (A. C. Newell, editor) Nonlinear Wave Motion, Amer. Math. Soc., 1974, pp. 215-216.

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Additional Information

Helge Holden
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway, — and — Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway

Christian Lubich
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany

Nils Henrik Risebro
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway

Keywords: Operator splitting, Burgers equation, KdV equation, Benney–Lin equation, Kawahara equation
Received by editor(s): February 8, 2011
Published electronically: June 12, 2012
Additional Notes: This work was supported in part by the Research Council of Norway.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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