A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

  • Authors:

  • Jae-Hun Jung

  • Affiliations:

  • Department of Mathematics, University of Massachusetts at Dartmouth, North Dartmouth, USA 02747-2300

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2009

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Abstract

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H c (x) on the collocation points is used for the approximation of singular source terms δ(x-c) or δ (n)(x-c) without any regularization. The direct projection of H c (x) yields highly oscillatory approximations of δ(x-c) and δ (n)(x-c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided.