A Dynamical Pseudo-Spectral Domain Decomposition Technique: Application to Viscous Compressible Flows

Abstract

A dynamical spectral domain decomposition method is presented. In each subdomain a transformation of coordinate is used. Both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theHω2-norm of the calculated solution. We show on some functions that the total norm of the Chebyshev series depends on the location of the interfaces. Moreover, there exists a minimum that defines the best location of the interface. This defines a dynamical generation of Chebyshev collocation points. This numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation. This algorithm is then used for the numerical solution of the time-dependent full Navier–Stokes equations. The solution technique consists in a Fourier–Chebyshev collocation method combined with a matching method. The computational domain is decomposed into subdomains in the vertical direction. In each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined. The elliptic problems coming from the viscous terms are solved by means of the Chebyshev acceleration method. Density is matched with an upwind procedure whereas the velocities, the temperature, and the concentration are handled with the influence matrix method. Numerical examples are carried out on the compressible Kelvin–Helmholtz and Rayleigh–Taylor flows.