Commun. Comput. Phys., 14 (2013), pp. 574-598.


Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

Hermann Brunner 1, Hongwei Li 2*, Xiaonan Wu 3

1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong; Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada.
2 College of Mathematical Sciences, Shandong Normal University, Jinan, 250014, P.R. China; Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.
3 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong.

Received 16 April 2012; Accepted (in revised version) 11 October 2012
Available online 25 January 2013
doi:10.4208/cicp.160412.111012a

Abstract

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.

AMS subject classifications: 65M06, 65M20

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Key words: Finite-time blow-up, nonlinear wave equation, absorbing boundary conditions, finite difference method, unbounded domains.

*Corresponding author.
Email: hbrunner@math.hkbu.edu.hk, hermann@math.mun.ca (H. Brunner), lhwqust@gmail.com (H. Li), xwu@hkbu.edu.hk (X. Wu)
 

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