Quadrature Imposition of Compatability Conditions in Chebyshev Methods
Abstract
Often, in solving a elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula can not be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula, however we show in this paper that this is the wrong choice and it may lead to divergence if time dependent methods are used to march the solution to steady state. We develop, in this paper, the correct quadrature formula for these problems. This formula takes into account the degree of the polynomials involved. We show that this formula leads to a well conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 1990
- Accession Number
- ADA228695
Entities
People
- C. L. Streett
- D. Gottlieb