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Finite-Element Solution of Nonlinear Time-Dependent Exterior Wave Problems

✍ Scribed by Dan Givoli; Igor Patlashenko

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
288 KB
Volume
143
Category
Article
ISSN
0021-9991

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✦ Synopsis

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 method. The governing equation is first discretized in time, leading to a time-stepping scheme, where a nonlinear exterior elliptic problem has to be solved in each time step. An artificial boundary B is introduced, which bounds the computational domain , and a simple-iteration procedure is used to linearize the problem in the infinite domain outside 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 . The computational aspects of this method are discussed. Numerical results are presented for the nonlinear wave equation, whose solutions may blow up in a finite time under certain conditions, and it is shown that the behavior of the solution predicted by theory is captured by the scheme.

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