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Center for Nonlinear Studies |
When periodic boundary conditions arise in fluid problems, spectral methods using Fourier discreitzations are often utilized because their high accuracy and efficiency. This efficiency is maintained in higher dimensional problems as long boundary conditions remain periodic in each additional direction. When periodicity is lost, as occurs in bounded domains for fluid problems, Chebyshev discretizations provide for spectral ac- curacy and the enforcement of more complicated boundary conditions. While it well known that differential equations in 1 dimension can be solved efficiently with Chebyshev discretizations, O(N) operations for N unknowns, this efficiency is lost in higher dimensions due to the coupling between Chebyshev modes. In this talk, I will present a new methodology, the “pseudo-inverse“ technique (PIT), for solving differential equations in multiple dimensions which is very efficient, O(N 3 ) operations for N 2 unknowns. This new methodology is compared to the matrix diagonalization technique (MDT) of Haidvogel [79], Shen [95], and Doha [06]. While the cost for MDT and PIT are the same in 2 dimensions, there are significant differences. MDT utilizes an eigenvalue/eigenvector decomposition and can only be used for relatively simple differential equations. PIT is based upon intrinsic properties of the Chebyshev polynomials and is adaptable to almost any PDE. In this talk I will introduce PIT, present results for a standard suite of test problems, and discuss of the adaptability of PIT to more complicated problems.