Solomonoff, A. | Search NASA

NASA NTRS1989

Global properties of pseudospectral methods

The present application of polynomial interpolation methods to function-approximation and numerical solutions for hyperbolic and elliptic PDEs allows the explicit construction of the derivative matrix for a general sequence of collocation points. An evaluation of the effect of several factors on the performance of these methods indicates an inability to interpret global methods in terms of local ones; the accuracy of the approximation will differ when the function's large gradients occur near the center of the region or near the boundary, irrespective of the boundary vicinity's collocation-point density.

Solomonoff, A.↗

NASA NTRS1986

Global collocation methods for approximation and the solution of partial differential equations

Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

Solomonoff, A.↗