Operator splitting for partial differential equations with Burgers nonlinearity

Holden, Helge; Lubich, Christian; Risebro, Nils

doi:10.1090/S0025-5718-2012-02624-X

Operator splitting for partial differential equations with Burgers nonlinearity
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by Helge Holden, Christian Lubich and Nils Henrik Risebro PDF

Math. Comp. 82 (2013), 173-185 Request permission

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 $r\ge 1$.

References

A. Ambrosetti and G. Prodi. A Primer of Nonlinear Analysis. Cambridge UP, Cambridge, 1995.
D. J. Benney, Long waves on liquid films, J. Math. and Phys. 45 (1966), 150–155. MR 201125
H. A. Biagioni and F. Linares, On the Benney-Lin and Kawahara equations, J. Math. Anal. Appl. 211 (1997), no. 1, 131–152. MR 1460163, DOI 10.1006/jmaa.1997.5438
J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR 1227985
Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie, and Nils Henrik Risebro, Splitting methods for partial differential equations with rough solutions, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010. Analysis and MATLAB programs. MR 2662342, DOI 10.4171/078
Helge Holden, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro, Operator splitting methods for generalized Korteweg-de Vries equations, J. Comput. Phys. 153 (1999), no. 1, 203–222. MR 1703652, DOI 10.1006/jcph.1999.6273
Helge Holden, Kenneth H. Karlsen, Nils Henrik Risebro, and Terence Tao, Operator splitting for the KdV equation, Math. Comp. 80 (2011), no. 274, 821–846. MR 2772097, DOI 10.1090/S0025-5718-2010-02402-0
H. Holden, K. H. Karlsen, T. Karper. Operator splitting for two-dimensional incompressible fluid equations. Math. Comp., to appear.
Tobias Jahnke and Christian Lubich, Error bounds for exponential operator splittings, BIT 40 (2000), no. 4, 735–744. MR 1799313, DOI 10.1023/A:1022396519656
Yueling Jia and Zhaohui Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations 246 (2009), no. 6, 2448–2467. MR 2498848, DOI 10.1016/j.jde.2008.10.027
T. Kawahara. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33:260–264 (1975).
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, 323–347. MR 1086966, DOI 10.1090/S0894-0347-1991-1086966-0
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
S. P. Lin. Finite amplitude side-band stability of a viscous film. J. Fluid. Mech. 63:417–429 (1974).
Felipe Linares and Gustavo Ponce, Introduction to nonlinear dispersive equations, Universitext, Springer, New York, 2009. MR 2492151
Christian Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), no. 264, 2141–2153. MR 2429878, DOI 10.1090/S0025-5718-08-02101-7
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
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.