FINITE-ELEMENT SOLUTION OF NONLINEAR TIME-DEPENDENT EXTERIOR WAVE PROBLEMS

A finite element scheme is devised for the solution of nonlinear time- dependent exterior wave problems. The two-dimensional nonlinear scalar (Klein-Gordon) wave equation is taken as a model to illustrate the me thod. The governing equation is first discretized in time, leading to a time-stepping scheme, where a nonlinear exterior elliptic problem ha s to be solved in each time step. An artificial boundary B is introduc ed, which bounds the computational domain Ohm, and a simple-iteration procedure is used to linearize the problem in the infinite domain outs ide B. This enables the derivation of a Dirichlet-to-Neumann boundary condition on B. Finite element discretization and Newton iteration are finally employed to solve the problem in Ohm. The computational aspec ts of this method are discussed. Numerical results are presented for t he nonlinear wave equation, whose solutions may blow up in a finite ti me under certain conditions, and it is shown that the behavior of the solution predicted by theory is captured by the scheme. (C) 1998 Acade mic Press.