Bubble stabilization of chebyshev spectral method for advection-diffusion equation

Abstract

It is well known that the spectral method is very accurate for a numerical approximation to the smooth solution of a boundary value problem. The spectral method, however, applied to the advection-diffusion equation is unstable when the diffusion coefficient is very small. For certain special types of advection-diffusion equations, Canuto and Puppo suggested a new scheme using the locally supported bubble functions to stabilize the Legendre spectral method. With the augmentation of locally supported bubble functions and a certain interpolation operator, two stabilization schemes are proposed to stabilize the Chebyshev spectral method for one and two dimensional advection-diffusion equations under certain assumptions for the velocity function. Theoretical analysis for the convergence and stability of the proposed schemes are presented. Numerical experiments for specific examples show an improvement of accuracy and stability.