Adaptive Pseudo-Spectral Domain Decomposition and the Approximation of Multiple Layers

Abstract

When Chebyshev pseudo-spectral methods are used with domain decomposition procedures in the numerical solution of partial differential equations, the use of multiple domains can significantly affect the accuracy of the approximation. This is particularly true when the solution exhibits layer type behavior, i.e., there are narrow regions of rapid variation. Accuracy may be enhanced if the interfaces between adjacent subdomains are such that large gradients occur near the interfaces, while accuracy can be degraded if the rapid variations occur in the interior of the subdomains. The use of appropriate mappings within each subdomain can improve the accuracy of the approximation by choosing mappings so that the transformed function is more readily approximated by a low order polynomial. The particular choice of mappings, however, depends critically on whether the solution exhibits boundary layer or interior layer behavior within each subdomain. We analyze the relationship between interface location and mappings required to obtain an efficient approximation of such functions. We compare two strategies, both based on constructing subdomains so that each subdomain contains only one layer. In the first strategy interface locations are chosen so that the rapid variations occur as interior layers and mappings are employed which enhance the resolution of such layers (strategy I for interior). In the second strategy interface locations are chosen so that rapid variations occur as boundary layers and mappings are employed which enhance resolution of boundary layers (strategy B for boundary). Both strategies lead to adaptive domain decomposition procedures based entirely on the locations of the layers. We demonstrate that strategy B offers superior accuracy for a given computational effort and employ this strategy in developing an adaptive domain decomposition method for problems with multiple layers. Both strategies are comparable regarding spectral radii of the resulting matrices, and we conclude that domain decomposition itself cannot result in larger stable time steps when accuracy of the approximation of the layer is considered. The adaptive domain decomposition method is illustrated by the computation of both axisymmetric and cellular flames with a sequential reaction mechanism involving two reaction zones.