Volume 1, Issue 6
Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

Jae-Hun Jung & Wai Sun Don

Adv. Appl. Math. Mech., 1 (2009), pp. 769-780.

Published online: 2009-01

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  • Abstract

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the δ-function. For the approximation of the δ-function, the direct projection method is used that was proposed in [6]. The δ-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The δ-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the nonstationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

  • Keywords

Singular sources Dirac-$\delta$-function Direct projection method Chebyshev collocation method WENO scheme

  • AMS Subject Headings

65M06 65M30 65M70

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COPYRIGHT: © Global Science Press

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@Article{AAMM-1-769, author = {Jae-Hun Jung and Wai Sun Don}, title = {Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {6}, pages = {769--780}, abstract = {

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the δ-function. For the approximation of the δ-function, the direct projection method is used that was proposed in [6]. The δ-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The δ-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the nonstationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m09S10}, url = {http://global-sci.org/intro/article_detail/aamm/8396.html} }
TY - JOUR T1 - Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources AU - Jae-Hun Jung & Wai Sun Don JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 769 EP - 780 PY - 2009 DA - 2009/01 SN - 1 DO - http://dor.org/10.4208/aamm.09-m09S10 UR - https://global-sci.org/intro/aamm/8396.html KW - Singular sources KW - Dirac-$\delta$-function KW - Direct projection method KW - Chebyshev collocation method KW - WENO scheme AB -

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the δ-function. For the approximation of the δ-function, the direct projection method is used that was proposed in [6]. The δ-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The δ-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the nonstationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

Jae-Hun Jung & Wai Sun Don. (1970). Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources. Advances in Applied Mathematics and Mechanics. 1 (6). 769-780. doi:10.4208/aamm.09-m09S10
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