Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

Year:    2009

Author:    Jae-Hun Jung, Wai Sun Don

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 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 non-stationary 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.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m09S10

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 6 : pp. 769–780

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    12

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

Author Details

Jae-Hun Jung

Wai Sun Don

  1. Dynamic response of bidirectional functionally graded beams with elastic supports and foundations under moving harmonic loads

    Chen, Wei-Ren | Lin, Chien-Hung

    Acta Mechanica, Vol. 235 (2024), Iss. 7 P.4833

    https://doi.org/10.1007/s00707-024-03975-2 [Citations: 1]
  2. Dynamic Analysis of Elastically Supported Functionally Graded Sandwich Beams Resting on Elastic Foundations Under Moving Loads

    Chen, Wei-Ren | Lin, Chien-Hung

    International Journal of Structural Stability and Dynamics, Vol. 24 (2024), Iss. 08

    https://doi.org/10.1142/S0219455424500871 [Citations: 0]
  3. Numerical Methods and Optimization

    Numerical Integration of Partial Differential Equations

    Corriou, Jean-Pierre

    2021

    https://doi.org/10.1007/978-3-030-89366-8_7 [Citations: 0]
  4. A High-Order Dirac-Delta Regularization with Optimal Scaling in the Spectral Solution of One-Dimensional Singular Hyperbolic Conservation Laws

    Suarez, Jean-Piero | Jacobs, Gustaaf B. | Don, Wai-Sun

    SIAM Journal on Scientific Computing, Vol. 36 (2014), Iss. 4 P.A1831

    https://doi.org/10.1137/130939341 [Citations: 19]
  5. A modified differential quadrature procedure for numerical solution of moving load problem

    Eftekhari, SA

    Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 230 (2016), Iss. 5 P.715

    https://doi.org/10.1177/0954406215584630 [Citations: 10]
  6. Detailed comparison of numerical methods for the perturbed sine-Gordon equation with impulsive forcing

    Wang, Danhua | Jung, Jae-Hun | Biondini, Gino

    Journal of Engineering Mathematics, Vol. 87 (2014), Iss. 1 P.167

    https://doi.org/10.1007/s10665-013-9678-x [Citations: 13]
  7. A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem

    Eftekhari, S.A.

    Latin American Journal of Solids and Structures, Vol. 13 (2016), Iss. 9 P.1763

    https://doi.org/10.1590/1679-78252251 [Citations: 8]
  8. An accurate differential quadrature procedure for the numerical solution of the moving load problem

    Eftekhari, S. A.

    Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 42 (2020), Iss. 5

    https://doi.org/10.1007/s40430-020-2247-0 [Citations: 1]
  9. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws

    Castro, Marcos | Costa, Bruno | Don, Wai Sun

    Journal of Computational Physics, Vol. 230 (2011), Iss. 5 P.1766

    https://doi.org/10.1016/j.jcp.2010.11.028 [Citations: 410]
  10. A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

    Eftekhari, S. A.

    Latin American Journal of Solids and Structures, Vol. 12 (2015), Iss. 7 P.1241

    https://doi.org/10.1590/1679-78251417 [Citations: 36]
  11. Efficient determination of the critical parameters and the statistical quantities for Klein–Gordon and sine-Gordon equations with a singular potential using generalized polynomial chaos methods

    Chakraborty, Debananda | Jung, Jae-Hun

    Journal of Computational Science, Vol. 4 (2013), Iss. 1-2 P.46

    https://doi.org/10.1016/j.jocs.2012.04.002 [Citations: 6]