Chebyshev Spectral Methods
High-accuracy discretizations for non-periodic problems that represent solutions as sums of Chebyshev polynomials.
Spectral accuracy without periodicity
Spectral methods represent a solution as a sum of global basis functions and can achieve error that decreases faster than any power of the grid spacing (spectral or exponential convergence) for smooth solutions. Fourier bases require periodicity; for non-periodic problems on an interval, Chebyshev polynomials are the standard choice. They combine the accuracy of spectral methods with the ability to handle general boundary conditions.
Why Chebyshev points
Interpolating a function at equally spaced points causes the Runge phenomenon: wild oscillations near the endpoints. Chebyshev methods instead use points clustered near the ends of the interval (the Chebyshev-Gauss-Lobatto points), which suppress this instability and give well-conditioned interpolation. Remarkably, a Chebyshev series is just a Fourier cosine series in disguise, so the fast Fourier transform can evaluate Chebyshev expansions efficiently.
Differentiation matrices
Derivatives are computed by multiplying the vector of nodal values by a dense differentiation matrix, or equivalently by transforming to coefficient space, differentiating the series, and transforming back. Boundary conditions are imposed by modifying the rows of the differentiation matrix corresponding to boundary points. Because the differentiation matrices are dense, spectral methods trade a small number of highly accurate degrees of freedom against the sparsity of finite differences.
- Exponential convergence for smooth solutions
- Handles general (non-periodic) boundary conditions
- Points clustered at ends avoid Runge oscillations
- Dense operators, so best for moderate resolution with smooth fields
Use and limits
Chebyshev methods excel where the solution is smooth and high accuracy per degree of freedom is worth dense operators: stability eigenvalue problems, boundary-layer resolution, and one-dimensional or tensor-product geometries. They lose their edge when solutions have discontinuities (where Gibbs oscillations return) or on complex geometries, where finite-element or Fourier approaches (for periodic directions) are preferred. In plasma stability analysis, Chebyshev discretization in the radial direction is a common and accurate choice.