Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation

✍ Scribed by Lonny L Thompson; Runnong Huan

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 395 KB
• Volume: 187
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/s0045-7825(99)00114-0

✦ Synopsis

When solving the wave equation in in®nite regions using ®nite element methods, the domain must be truncated. We investigate the accuracy of time-dependent non-re¯ecting boundary conditions (NRBC) derived in Grote, Keller (1995), when implemented in the ®nite element method. The NRBC annihilate the ®rst N wave harmonics on a spherical truncation boundary. High-order temporal derivatives are formulated as a system of ®rst-order ordinary dierential equations. Several versions of implicit and explicit multi-step, time-integration schemes are presented for solution of the ®nite element equations concurrently with the ®rst-order system appearing in the NRBC. An alternative scaling of the boundary variables is introduced which leads to a well-conditioned coecient matrix. Although the boundary conditions are global over the boundary, when implemented in the ®nite element method, they only require inner products of spherical harmonics within the force vector, and as a result, they are easy to implement and do not disturb the banded/ sparse structure of the matrix equations. Several numerical examples are presented which demonstrate the improvement in accuracy over standard ®nite element methods.