  Singular source terms expressed as weighted summations of Dirac-delta
functions are regularized through approximation theory with convolution
operators. We consider the numerical solution of scalar and one-dimensional
hyperbolic conservation laws with the singular source by spectral Chebyshev
collocation methods. The regularization is obtained by convolution with a
high-order compactly supported Dirac-delta approximation whose overall accuracy
is controlled by the number of vanishing moments, degree of smoothness and
length of the support (scaling parameter). An optimal scaling parameter that
leads to a high-order accurate representation of the singular source at smooth
parts and full convergence order away from the singularities in the spectral
solution is derived. The accuracy of the regularization and the spectral
solution is assessed by solving an advection and Burgers equation with smooth
initial data. Nu- merical results illustrate the enhanced accuracy of the
spectral method through the proposed regularization.
