A penalty approach to the numerical simulation of a constrained wave motion

Abstract — The main goal of this article is to investigate the numerical solution of a vector-valued nonlinear wave equation, the nonlinearity being of the Ginzburg-Landau type, namely (|~u|2 − 1)~u. This equation is obtained when treating by penalty a constrained wave-motion, where the displacement vector is of constant length (1 here, after rescaling). An important step of the approximation process is the construction of a time discretization scheme preserving in some sense the energy conservation property of the continuous model.The stability properties of the above scheme are discussed. The authors discuss also the finite element approximation and the quasi-Newton solution of the nonlinear elliptic system obtained at each time step from the time discretization. The results of numerical experiments are presented; they show that for the constraint of the original wave problem to be accurately verified we need to use a small value of the penalty parameter.