High order numerical methods for differential equations with singular sources

Applied and Computational Mathematics Seminar
Monday, April 19, 2010 - 13:00
1 hour (actually 50 minutes)
Skiles 255
Mathematics, SUNY Buffalo
Solutions of differential equations with singular source terms easily becomenon-smooth or even discontinuous. High order approximations of suchsolutions yield the Gibbs phenomenon. This results in the deterioration ofhigh order accuracy. If the problem is nonlinear and time-dependent it mayalso destroy the stability. In this presentation, we focus on thedevelopment of high order methods to obtain high order accuracy rather thanregularization methods. Regularization yields a good stability condition,but may lose the desired accuracy. We explain how high order collocationmethods can be used to enhance accuracy, for which we will adopt severalmethods including the Green’s function approach and the polynomial chaosmethod. We also present numerical issues associated with the collocationmethods. Numerical results will be presented for some differential equationsincluding the nonlinear sine-Gordon equation and the Zerilli equation.