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On the numerical solution to a nonlinear wave equation associated with the first Painlevé equation: an operator-splitting approach. (English) Zbl 1268.65112

Summary: 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 G. Strang’s symmetrized operator-splitting scheme [SIAM J. Numer. Anal. 5, 506–517 (1968; Zbl 0184.38503)]. 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
65L05 Numerical methods for initial value problems involving ordinary differential equations
35L70 Second-order nonlinear hyperbolic equations

Citations:

Zbl 0184.38503

Software:

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References:

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