Collocation methods for hyperbolic partial differential equations with singular sources

Jae Hun Jung*, Wai Sun DON

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)769-780
Number of pages12
JournalAdvances in Applied Mathematics and Mechanics
Volume1
Issue number6
DOIs
Publication statusPublished - 2009

Scopus Subject Areas

  • Mechanical Engineering
  • Applied Mathematics

User-Defined Keywords

  • Chebyshev collocation method
  • Dirac-δ-function
  • Direct projection method
  • Singular sources
  • WENO scheme

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