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Volume 24, Issue 3
Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations

Bradley E. Treeby, Elliott S. Wise & B. T. Cox

DOI: 10.4208/cicp.OA-2017-0192

Commun. Comput. Phys., 24 (2018), pp. 623-634.

Published online: 2018-05

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  • Abstract

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.

  • Keywords

Nonstandard finite difference, pseudospectral time domain, PSTD, Fourier collocation spectral method, wave equation, diffusion.

  • AMS Subject Headings

65M70, 65M06, 65M80

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-623, author = {Bradley E. Treeby, Elliott S. Wise and B. T. Cox}, title = {Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {3}, pages = {623--634}, abstract = {

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0192}, url = {http://global-sci.org/intro/article_detail/cicp/12273.html} }
TY - JOUR T1 - Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations AU - Bradley E. Treeby, Elliott S. Wise & B. T. Cox JO - Communications in Computational Physics VL - 3 SP - 623 EP - 634 PY - 2018 DA - 2018/05 SN - 24 DO - http://doi.org/10.4208/cicp.OA-2017-0192 UR - https://global-sci.org/intro/article_detail/cicp/12273.html KW - Nonstandard finite difference, pseudospectral time domain, PSTD, Fourier collocation spectral method, wave equation, diffusion. AB -

A class of nonstandard pseudospectral time domain (PSTD) schemes for solving time-dependent hyperbolic and parabolic partial differential equations (PDEs) is introduced. These schemes use the Fourier collocation spectral method to compute spatial gradients and a nonstandard finite difference scheme to integrate forwards in time. The modified denominator function that makes the finite difference time scheme exact is transformed into the spatial frequency domain or k-space using the dispersion relation for the governing PDE. This allows the correction factor to be applied in the spatial frequency domain as part of the spatial gradient calculation. The derived schemes can be formulated to be unconditionally stable, and apply to PDEs in any space dimension. Examples of the resulting nonstandard PSTD schemes for several PDEs are given, including the wave equation, diffusion equation, and convection-diffusion equation.

Bradley E. Treeby, Elliott S. Wise and B. T. Cox. (2018). Nonstandard Fourier Pseudospectral Time Domain (PSTD) Schemes for Partial Differential Equations. Communications in Computational Physics. 24 (3). 623-634. doi:10.4208/cicp.OA-2017-0192
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