On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlevé Equation: An Operator-Splitting Approach
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.
KeywordsPainlevé equation Nonlinear wave equation Blow-up solution Operator-splitting
Mathematics Subject Classification35L70 65N06
The authors would like to thank the founders of the Painlevé Project. Their one page article in the Notices of the AMS  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.
- 1.Bornemann, F., Clarkson, P., Deift, P., et al.: A request: The Painlevé Project. Not. Am. Math. Soc. 57(11), 1938 (2010) Google Scholar
- 4.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) Google Scholar
- 7.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) Google Scholar
- 11.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) MathSciNetCrossRefzbMATHGoogle Scholar
- 15.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) Google Scholar
- 16.Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis. AMS, Providence (2001) Google Scholar