12. Advection-Diffusion Equation
Society for Industrial and Applied Mathematics eBooks · 1977
In this section, we consider spectral methods for the advection-diffusion (“linearized Burgers”) equation ∂u (x,t) ∂t +U ∂u (x,t) ∂x =ν ∂2 u ∂ x2 +ƒ (x,t) ,−1≦x≦1, 12.1 u (−1,t)=0,u (1,t)=0 , 12.2 u (x,0)=g (x). 12.3 Equation (12.1) is parabolic so boundary conditions should be applied at both and . When ν is small, the boundary condition applied at (assuming ) has an interesting effect on the stability of the spectral methods. To begin, we remark that the analyses of §§ 7–8 can be extended to show that, as N→∞, N-term Legendre and Chebyshev approximations to (12.1)–(12.3) are stable and convergent.