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 .
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.