Quadrature imposition of compatibility conditions in Chebyshev methods
Abstract
Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for wellposedness. 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 GaussChebyshev (or any variant of it, like the GaussLobatto) 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 ClenshawCurtis formula, however it is shown 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. The correct quadrature formula is developed for these problems. This formula takes into account the degree of the polynomials involved. It is shown that this formula leads to a well conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.
 Publication:

Final Report Institute for Computer Applications in Science and Engineering
 Pub Date:
 October 1990
 Bibcode:
 1990icas.reptR....G
 Keywords:

 Chebyshev Approximation;
 Integrals;
 Neumann Problem;
 Quadratures;
 Compatibility;
 Divergence;
 Partial Differential Equations;
 Polynomials;
 Spectral Methods;
 Steady State;
 Fluid Mechanics and Heat Transfer