On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev Equation:an Operator-Splitting Approach  

作　　者：Roland GLOWINSKI Annalisa QUAINI 

机构地区：[1]Department of Mathematics,University of Houston,4800 Calhoun Rd,Houston(TX) 77204,USA [2]Laboratoire Jacques-Louis Lions,Universit Pierre et Marie Curie,4,Place Jussieu,75252 Paris CEDEX 05,France

出　　处：《Chinese Annals of Mathematics,Series B》2013年第2期237-254,共18页数学年刊（B辑英文版）

摘　　要：The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevd 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 Painlevd 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.

关 键 词：Painlevd equation Nonlinear wave equation Blow-up solution Operator-SDlitting 

分 类 号：O241.8[理学—计算数学]