Spectral methods based on nonclassical basis functions: the advection-diffusion equation

Abstract The advection–diffusion equation has a long history as a benchmark for numerical methods. If advection dominates over diffusion the numerical solution is difficult, especially if there are boundary layers to resolve. The eigenvalues of the approximate discretized spatial operator can be complex, and if the real part of any one of these is positive, the temporal development of the discretized equations by finite differences is unstable. The stability of the time development by finite difference methods is usually discussed in terms of the eigenvalues of the first and second derivative spatial operators. In this paper, the eigenvalues of the spatial operator in the advection–diffusion equation determined with a Galerkin method based on a new set of nonclassical basis functions are all real and negative. A collocation solution of the time dependent advection–diffusion equation is also considered and results using Chebyshev–Gauss–Lobatto and Legendre–Gauss–Lobatto quadratures are compared with results based on new basis functions. The results demonstrate that improved convergence can be obtained with new basis functions defined with respect to nonclassical weight functions.

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