Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods

Abstract

A robust methodology is presented for efficiently solving partial differential equations using Chebyshev spectral techniques. It is well known that differential equations in one dimension can be solved efficiently with Chebyshev discretizations, O( N) operations for N unknowns, however this efficiency is lost in higher dimensions due to the coupling between modes. This paper presents the “quasi-inverse“ technique (QIT), which combines optimizations of one-dimensional spectral differentiation matrices with Kronecker matrix products to build efficient multi-dimensional operators. This strategy results in O( N 2 D−1 ) operations for N D unknowns, independent of the form of the differential operators. QIT is compared to the matrix diagonalization technique (MDT) of Haidvogel and Zang [D.B. Haidvogel, T. Zang, The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. Comput. Phys. 30 (1979) 167–180] and Shen [J. Shen, Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comp. 16 (1) (1995) 74–87]. While the cost for MDT and QIT are the same in two dimensions, there are significant differences. MDT utilizes an eigenvalue/eigenvector decomposition and can only be used for relatively simple differential equations. QIT is based upon intrinsic properties of the Chebyshev polynomials and is adaptable to linear PDEs with constant coefficients in simple domains. We present results for a standard suite of test problems, and discuss of the adaptability of QIT to more complicated problems.