Fourier Galerkin Approach to Wave Equation with Absorbing Boundary Conditions

Authors: Alexandra Leukauf, Alexander Schirrer, Emir Talic

Abstract:

Numerical computation of wave propagation in a large domain usually requires significant computational effort. Hence, the considered domain must be truncated to a smaller domain of interest. In addition, special boundary conditions, which absorb the outward travelling waves, need to be implemented in order to describe the system domains correctly. In this work, the linear one dimensional wave equation is approximated by utilizing the Fourier Galerkin approach. Furthermore, the artificial boundaries are realized with absorbing boundary conditions. Within this work, a systematic work flow for setting up the wave problem, including the absorbing boundary conditions, is proposed. As a result, a convenient modal system description with an effective absorbing boundary formulation is established. Moreover, the truncated model shows high accuracy compared to the global domain.

Keywords: Absorbing boundary conditions, boundary control, Fourier Galerkin approach, modal approach, wave equation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1131844

References:

[1] R. Clayton and B. Engquist, “Absorbing boundary conditions for acoustic and elastic wave equations,” Bulletin of the Seismological Society of America, vol. 67, no. 6, pp. 1529–1540, 1977.
[Online]. Available: http://www.bssaonline.org/content/67/6/1529.abstract
[2] G. Aschauer, A. Schirrer, and S.-M. Jakubek, “Realtime-capable finite element model of railway catenary dynamics in moving coordinates,” IEEE Multi-Conference on Systems and Control, 2016.
[3] A. Facchinetti, L. Gasparetto, and S. Bruni, “Real-time catenary models for the hardware-in-the-loop simulation of the pantograph–catenary interaction,” Vehicle System Dynamics, vol. 51, no. 4, pp. 499–516, 2013.
[4] B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proceedings of the National Academy of Sciences, vol. 74, no. 5, pp. 1765–1766, 1977.
[Online]. Available: http://www.pnas.org/content/74/5/1765.abstract
[5] W. Guo and Z.-C. Shao, “Strong stability of an unstable wave equation by boundary feedback with only displacement observation,” IEEE Transactions on Automatic Control, vol. 57, no. 9, pp. 2367–2372, 2012.
[6] C. Cerjan, D. Kosloff, R. Kosloff, and M. Reshef, “A nonreflecting boundary condition for discrete acoustic and elastic wave equations,” Geophysics, vol. 50, no. 4, pp. 705–708, 1985.
[7] J. Sochacki, R. Kubichek, J. George, W.-R. Fletcher, and S. Smithson, “Absorbing boundary conditions and surface waves,” Geophysics, vol. 52, no. 1, pp. 60–71, 1987.
[8] B. Chen, D.-G. Fang, and B.-H. Zhou, “Modified Berenger PML absorbing boundary condition for FD-TD meshes,” IEEE Microwave and Guided Wave Letters, vol. 5, no. 11, pp. 399–401, 1995.
[9] D. Komatitsch and J. Tromp, “A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation,” Geophysical Journal International, vol. 154, no. 1, pp. 146–153, 2003.
[10] J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of computational physics, vol. 114, no. 2, pp. 185–200, 1994.
[11] W. A. Strauss, Partial differential equations - An introduction, second edition ed.