Quasi‐Uniform Spectral Schemes (QUSS), Part I: Constructing Generalized Ellipses for Graphical Grid Generation

Abstract

Chebyshev and Legendre polynomial spectral methods are bedeviled by highly nonuniform grids. The separation between nearest neighbors of an N -point grid at the center of the interval is π/2 larger than the spacing of a uniform grid with the same number of points. Quasi-Uniform Spectral Schemes (QUSS) redistribute grid points and choose basis functions in order to recover this factor of π/2 as nearly as possible while retaining a high density of points near the endpoints to avoid the horrors of the Gibbs or Runge Phenomenon. Here, we introduce a systematic approach, dubbed "mapped cosine bases," that embraces the widely used Kosloff/Tal-Ezer functions as a special case. The mapped cosine approach uses grid points x j that are the images of a uniform grid under the coordinate mapping x = τ inv (t). Here, we show how to generalize the well-known graphical construction of the Chebyshev grid using a circle to QUSS mappings using a generalized ellipse. This provides a way to visualize the maps and grids and the subtle differences between different mappings of the mapped cosine family. We illustrate and compare the Kosloff/Tal-Ezer map with two new maps that use elliptic integrals and Jacobian theta functions, respectively. We show that the elliptic integral grid is an asymptotic approximation to the usual grid for prolate spheroidal functions. This suggests the conjecture that one can obtain the benefits of a prolate basis without the complications of prolate functions by using mapped polynomials instead.

AAA interpolation of equispaced data

“…The general conformal mapping point of view was put forward in [29] and in [47, chapter 22]. See also [12].…”

AAA interpolation of equispaced data

“…The general conformal mapping point of view was put forward in [27] and in [44, chapter 22]. See also [11].…”