Truncation error in grid generation: A case study

Abstract

Theoretical proofs state that the planar Winslow or homogenous Thompson-Thames-Mastin (hTTM) map is a diffeomorphism, yet numerical solutions to the hTTM equations produce folded grids on the so-called "horseshoe" domain. A quasi-analytic solution to the horseshoe problem is constructed to demonstrate that folding is due to truncation error effects. Higher-order difference methods are also explored. 0 1995 John Wiley & Sons. Inc.

On 2D structured mesh generation by using mappings

“…In Ref. [5], it was supposed that this was caused by the truncation error. The analytical solution, presented in Fig.…”

“…[4] and [5], an example of the domain was considered, where the use of Eqs. (8) and (9) other two sides are the straight line segments, see Fig.…”

“…In Ref. [5], it was supposed that grid folding was due to the truncation error. A detail analysis of this issue was not performed as a search of an analytical solution of the Dirichlet problem for the Laplace equation on a domain of complex geometry is a very difficult task.…”

On a Problem of the Constructive Theory of Harmonic Mappings

“…In this situation, there often appears the effect of an "irremovable" error, that is, neither the growth of approximation order of the difference scheme nor refining of the grid result in substantial reduction of the error. A typical problem of such kind is the problem of a harmonic mapping of a "horseshoe-like" domain onto a square, well-known from papers [6,44,49,71]. We will call it the Roache-Steinberg problem, from the names of the authors of the pioneering paper [71].…”

“…The solution of the Roache-Steinberg problem with relative precision 10 −9 obtained in this paper by the multipole method is used as a test for a detailed study of the effect of "irremovable" error, which appeared in papers [6,49,71] in the course of solving this problem by difference methods based on Winslow's approach.…”