ADI spectral collocation methods for parabolic problems

• Published in: Journal of computational physics Vol. 229; no. 13; pp. 5182 - 5193
• Main Authors: Bialecki, B., de Frutos, J.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.07.2010; Elsevier
• Subjects: ADI; Chebyshev approximation; Coefficients; Collocation methods; Computational techniques; Crank–Nicolson; Dirichlet problem; Exact sciences and technology; Laplace modified; Legendre and Chebyshev polynomials; Mathematical analysis; Mathematical methods in physics; Mathematical models; Parabolic initial-boundary value problems; Physics; Spectra; Spectral collocation; Spectral lines; Crank–Nicolson; Legendre and Chebyshev polynomials; Parabolic initial-boundary value problems; Spectral collocation; ADI; Laplace modified; Second order; Band structure; Spectral method; Heat equation; Chebyshev polynomial; Boundary conditions; Calculation methods; Crank-Nicolson; Fast Fourier transforms; Discretization; Dirichlet problem; Boundary-value problems; Initial value problems; Calculation; Legendre polynomials
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.03.033

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.