Research PaperNo Access

Pseudospectral Time-Domain (PSTD) Methods for the Wave Equation: Realizing Boundary Conditions with Discrete Sine and Cosine Transforms

Elliott S. Wise

Department of Medical Physics and Biomedical Engineering, University College London, UK

Search for more papers by this author

Jiri Jaros

Centre of Excellence IT4Innovations, Faculty of Information Technology, Brno University of Technology, Czech Republic

Search for more papers by this author

Ben T. Cox

Department of Medical Physics and Biomedical Engineering, University College London, UK

Search for more papers by this author

, and

Bradley E. Treeby

https://orcid.org/0000-0001-7782-011X

Department of Medical Physics and Biomedical Engineering, University College London, UK

E-mail Address: b.treeby@ucl.ac.uk

Search for more papers by this author

https://doi.org/10.1142/S2591728520500218Cited by:4 (Source: Crossref)

Abstract

Pseudospectral time domain (PSTD) methods are widely used in many branches of acoustics for the numerical solution of the wave equation, including biomedical ultrasound and seismology. The use of the Fourier collocation spectral method in particular has many computational advantages. However, the use of a discrete Fourier basis is also inherently restricted to solving problems with periodic boundary conditions. Here, a family of spectral collocation methods based on the use of a sine or cosine basis is described. These retain the computational advantages of the Fourier collocation method but instead allow homogeneous Dirichlet (sound-soft) and Neumann (sound-hard) boundary conditions to be imposed. The basis function weights are computed numerically using the discrete sine and cosine transforms, which can be implemented using $O (N log N)$ operations analogous to the fast Fourier transform. Practical details of how to implement spectral methods using discrete sine and cosine transforms are provided. The technique is then illustrated through the solution of the wave equation in a rectangular domain subject to different combinations of boundary conditions. The extension to boundaries with arbitrary real reflection coefficients or boundaries that are nonreflecting is also demonstrated using the weighted summation of the solutions with Dirichlet and Neumann boundary conditions.

Keywords:

References

1. B. Fornberg and D. M. Sloan , A review of pseudospectral methods for solving partial differential equations, Acta Numer. 3 (1994) 203–267. Crossref, Google Scholar
2. L. N. Trefethen , Spectral Methods in MATLAB (Siam, Philadelphia, 2000). Crossref, Google Scholar
3. Q. H. Liu , Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm, IEEE Trans. Geosci. Remote Sens. 37 (1999) 917–926. Crossref, Google Scholar
4. J. P. Boyd , Chebyshev and Fourier Spectral Methods (Dover Publications, Mineola, New York, 2001). Google Scholar
5. J. S. Hesthaven, S. Gottlieb and D. Gottlieb , Spectral Methods for Time-Dependent Problems, Vol. 21 (Cambridge University Press, London, 2007). Crossref, Google Scholar
6. T. D. Mast, L. P. Souriau, D.-L. Liu, M. Tabei, A. I. Nachman and R. C. Waag , A k-space method for large-scale models of wave propagation in tissue, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48 (2001) 341–354. Crossref, Google Scholar
7. J. L. Robertson, B. T. Cox, J. Jaros and B. E. Treeby , Accurate simulation of transcranial ultrasound propagation for ultrasonic neuromodulation and stimulation, J. Acoust. Soc. Am. 141 (2017) 1726–1738. Crossref, Google Scholar
8. B. Fornberg , The pseudospectral method: Comparisons with finite differences for the elastic wave equation, Geophysics 52 (1987) 483–501. Crossref, Google Scholar
9. N. Albin and O. P. Bruno , A spectral FC solver for the compressible Navier–Stokes equations in general domains I: Explicit time-stepping, J. Comput. Phys. 230 (2011) 6248–6270. Crossref, Google Scholar
10. B. E. Treeby, J. Jaros, A. P. Rendell and B. Cox , Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method, J. Acoust. Soc. Am. 131 (2012) 4324–4336. Crossref, Google Scholar
11. J.-P. Berenger , A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200. Crossref, Google Scholar
12. J.-P. Bérenger , Perfectly matched layer (PML) for computational electromagnetics, Synth. Lect. Comput. Electromagn. 2 (2007) 1–117. Crossref, Google Scholar
13. C. Spa, A. Garriga and J. Escolano , Impedance boundary conditions for pseudo-spectral time-domain methods in room acoustics, Appl. Acoust. 71 (2010) 402–410. Crossref, Google Scholar
14. C. Spa, J. Escolano and A. Garriga , Semi-empirical boundary conditions for the linearized acoustic euler equations using pseudo-spectral time-domain methods, Appl. Acoust. 72 (2011) 226–230. Crossref, Google Scholar
15. J. D. Jackson , Classical Electrodynamics (Wiley, New York, 1999). Google Scholar
16. L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders , Fundamentals of Acoustics (John Wiley and Sons, New York, 1999). Google Scholar
17. B. Cox and P. Beard , Photoacoustic tomography with a single detector in a reverberant cavity, J. Acoust. Soc. Am. 125 (2009) 1426–1436. Crossref, Google Scholar
18. G. Strang , The discrete cosine transform, SIAM Rev. 41 (1999) 135–147. Crossref, Google Scholar
19. D. Kosloff and R. Kosloff , A non-periodic Fourier method for solution of the classical wave equation, Comput. Phys. Commun. 30 (1983) 333–336. Crossref, Google Scholar
20. T. Furumura and H. Takenaka , 2.5D modelling of elastic waves using the pseudospectral method, Geophys. J. Int. 124 (1996) 820–832. Crossref, Google Scholar
21. W.-K. Leung and Y. Chen , Transformed-space nonuniform pseudospectral time-domain algorithm, Microw. Opt. Technol. Lett. 28 (2001) 391–396. Crossref, Google Scholar
22. M. W. Feise, J. B. Schneider and P. J. Bevelacqua , Finite-difference and pseudospectral time-domain methods applied to backward-wave metamaterials, IEEE Trans. Antennas Propag. 52 (2004) 2955–2962. Crossref, Google Scholar
23. A. Bueno-Orovio and V. M. Pérez-García , Spectral smoothed boundary methods: the role of external boundary conditions, Numer. Methods Partial Differ. Equ. Int. J. 22 (2006) 435–448. Crossref, Google Scholar
24. B. Long and E. Reinhard , Real-time fluid simulation using discrete sine/cosine transforms, in Proc. 2009 Symp. Interactive 3D graphics and games (2009), pp. 99–106. Crossref, Google Scholar
25. M. Van Reeuwijk, S. A. Mathias, C. T. Simmons and J. D. Ward, Insights from a pseudospectral approach to the Elder problem, Water Resour. Res. 45. Google Scholar
26. W. Bao, X. Dong and X. Zhao , An exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system, SIAM J. Sci. Comput. 35 (2013) A2903–A2927. Crossref, Google Scholar
27. I. Ito , Pseudo spectral method based on symmetric extension, in 2018 7th European Workshop Visual Information Processing (EUVIP) (IEEE, 2018), pp. 1–6. Crossref, Google Scholar
28. Z. Wang and B. Hunt , The discreteW transform, Appl. Math. Comput. 16 (1985) 19–48. Crossref, Google Scholar
29. S. A. Martucci , Symmetric convolution and the discrete sine and cosine transforms, IEEE Trans. Signal Process. 42 (1994) 1038–1051. Crossref, Google Scholar
30. B. Fornberg , A Practical Guide to Pseudospectral Methods (Cambridge University Press, London, 1998). Google Scholar
31. M. Tabei, T. D. Mast and R. C. Waag , A k-space method for coupled first-order acoustic propagation equations, J. Acoust. Soc. Amer. 111 (2002) 53–63. Crossref, Google Scholar
32. B. Fornberg , High-order finite differences and the pseudospectral method on staggered grids, SIAM J. Numer. Anal. 27 (1990) 904–918. Crossref, Google Scholar
33. H.-W. Chen , Staggered-grid pseudospectral viscoacoustic wave field simulation in two-dimensional media, J. Acoust. Soc. Am. 100 (1996) 120–131. Crossref, Google Scholar
34. G. Strang , Computational Science and Engineering (Wellesley-Cambridge Press, 2007). Google Scholar
35. G. Wojcik, B. Fomberg, R. Waag, L. Carcione, J. Mould, L. Nikodym and T. Driscoll , Pseudospectral methods for large-scale bioacoustic models, in 1997 IEEE Ultrasonics Symp. Proc. Vol. 2 (IEEE, 1997), pp. 1501–1506. Crossref, Google Scholar
36. J. Lu, J. Pan and B. Xu , Time-domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method, J. Acoust. Soc. Am. 118 (2005) 3408–3419. Crossref, Google Scholar
37. M. Hornikx, R. Waxler and J. Forssén , The extended Fourier pseudospectral time-domain method for atmospheric sound propagation, J. Acoust. Soc. Am. 128 (2010) 1632–1646. Crossref, Google Scholar
38. B. Treeby, E. Wise and B. Cox , Nonstandard Fourier pseudospectral time domain (PSTD) schemes for partial differential equations, Commun. Comput. Phys. 24 (2018) 623–634. Crossref, Google Scholar
39. J. Jaros, A. P. Rendell and B. E. Treeby , Full-wave nonlinear ultrasound simulation on distributed clusters with applications in high-intensity focused ultrasound, Int. J. High Perform. Comput. Appl. 30 (2016) 137–155. Crossref, Google Scholar
40. V. Britanak, P. C. Yip and K. R. Rao , Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations (Elsevier, Oxford, 2007). Crossref, Google Scholar
41. M. Frigo and S. G. Johnson, FFTW (User Manual), Massachusetts Institute of Technology. Google Scholar
42. R. Reeves and K. Kubik , Shift, scaling and derivative properties for the discrete cosine transform, Signal Process. 86 (2006) 1597–1603. Crossref, Google Scholar
43. M. Frigo and S. G. Johnson , The design and implementation of FFTW3, Proc. IEEE 93 (2005) 216–231. Crossref, Google Scholar
44. B. E. Treeby, Matlab discrete trigonometric transform library (2020) https://github.com/ucl-bug/matlab-dtts, Last accessed 25 April 2020. Google Scholar
45. W. D. Smith , A nonreflecting plane boundary for wave propagation problems, J. Comput. Phys. 15 (1974) 492–503. Crossref, Google Scholar