Abstract
The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bornemann, F., Clarkson, P., Deift, P., et al.: A request: The Painlevé Project. Not. Am. Math. Soc. 57(11), 1938 (2010)
Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci., Ser. A 18(3), 1137–1161 (1982)
Wong, R., Zhang, H.Y.: On the connection formulas of the fourth Painlevé transcendent. Anal. Appl. 7(4), 419–448 (2009)
Clarkson, P.A.: Painlevé transcendents. In: Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.) NIST Handbook of Mathematical Functions, pp. 723–740. Cambridge University Press, Cambridge (2010)
Fornberg, B., Weideman, J.A.C.: A numerical methodology for the Painlevé equations. J. Comput. Phys. 230(15), 5957–5973 (2011)
Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)
Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)
Bokil, V.A., Glowinski, R.: An operator-splitting scheme with a distributed Lagrange multiplier based fictitious domain method for wave propagation problems. J. Comput. Phys. 205(1), 242–268 (2005)
Glowinski, R., Shiau, L., Sheppard, M.: Numerical methods for a class of nonlinear integro-differential equations. Calcolo 50, 17–33 (2013)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)
Glowinski, R., Dean, E.J., Guidoboni, G., et al.: Application of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution to the two-dimensional elliptic Monge-Ampère equation. Jpn. J. Ind. Appl. Math. 25(1), 1–63 (2008)
Chorin, A.J., Hughes, T.J.R., McCracken, M.F., et al.: Product formulas and numerical algorithms. Commun. Pure Appl. Math. 31, 205–256 (1978)
Beale, J.T., Majda, A.: Rates of convergence for viscous splitting of the Navier-Stokes equations. Math. Comput. 37(156), 243–259 (1981)
Leveque, R.J., Oliger, J.: Numerical methods based on additive splittings for hyperbolic partial differential equations. Math. Comput. 40(162), 469–497 (1983)
Marchuk, G.I.: Splitting and alternating direction method. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. I, pp. 197–462. North-Holland, Amsterdam (1990)
Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS, Providence (2001)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Keller, J.B.: On solutions of nonlinear wave equations. Commun. Pure Appl. Math. 10(4), 523–530 (1957)
Acknowledgements
The authors would like to thank the founders of the Painlevé Project. Their one page article in the Notices of the AMS [1] was the motivation of the work reported in this article. The authors thank Dr. Simone Paulotto for his help on the spectral analysis of some results.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Glowinski, R., Quaini, A. (2014). On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlevé Equation: An Operator-Splitting Approach. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-41401-5_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41400-8
Online ISBN: 978-3-642-41401-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)