Methods and Applications of Analysis

Volume 20 (2013)

Number 4

Special issue dedicated to the 70th birthday of Stanley Osher: Part I

Guest Editor: Chi-Wang Shu, Brown University

When Euler-Poisson-Darboux meets Painlevé and Bratu: On the numerical solution of nonlinear wave equations

Pages: 405 – 424



Roland Glowinski (Department of Mathematics, University of Houston, Texas, U.S.A.; Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France)

Annalisa Quaini (Department of Mathematics, University of Houston, Texas, U.S.A.)


The main goal of this article is to extend to Euler-Poisson-Darboux nonlinear wave equations the computational methods we employed in a previous work to solve a nonlinear equation coupling the classical wave operator with the nonlinear forcing term of the Painlevé I ordinary differential equation. In order to handle the extra (dissipative) term with singular coefficient encountered in the Euler-Poisson-Darboux equations, we advocate a five stage symmetrized operator-splitting scheme for the time-discretization. This scheme, combined with a finite element space discretization and adaptive time-stepping to monitor possible blow-up of the solution, provides a robust and accurate solution methodology, as shown by the results of the numerical experiments reported here. The nonlinearities we have considered are those encountered in the Painlevé I and II equations (and close variants of them), and the exponential one encountered in the celebrated Bratu problem.


Euler-Poisson-Darboux nonlinear wave equations, Painlevé equations, Bratu problem, blow-up solutions, operator-splitting

2010 Mathematics Subject Classification

35L70, 65N30

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