Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources (Advances in Applied Mathematics and Mechanics, 2021)
A numerical study is given on the spectral methods and the high orderWENO finite difference scheme for the solution of linear and nonlinear hyperbolicpartial differential equations with stationary and non-stationary singular sources.The singular source term is represented by the $δ$-function. For the approximationof 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. Nonlinearsine-Gordon equation with a stationary singular source was solved with theChebyshev collocation method. The $δ$-function with the spectral method is highlyoscillatory but yields good results with small number of collocation points. Theresults are compared with those computed by the second order finite differencemethod. In modeling general hyperbolic equations with a non-stationary singularsource, however, the solution of the linear scalar wave equation with the non-stationarysingular source using the direct projection method yields non-physicaloscillations for both the spectral method and the WENO scheme. The numericalartifacts arising when the non-stationary singular source term is considered on thediscrete grids are explained.