Acceleration of Convergence and Spectrum Transformation of Implicit Finite Difference Operators Associated with Navier-Stokes Equations

Abstract

Eigensystem analysis techniques are applied to finite difference formulations of Euler and Navier-Stokes equations in two dimensions. Spectrums of the resulting implicit difference operators are computed. The convergence and stability properties of the iterative methods are studied by making into account, the effect of grid geometry, time-step, numerical dissipation,, viscosity, boundary conditions, and the physics of the underlying flow. The largest eigenvalues are computed by using the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the dominant eigenvalues with the rate of convergence of the iterative method. Based on the pattern of eigenvalues distributions for various flow configurations, the feasability of applying existing convergence-acceleration techniques like eigenvalue annihilation and relaxation are discussed. Finally a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. The procedure of shifting the spectrum is applied to ARC2D, a flow code developed and being used at NASA Ames Research Center. When compared to eigenvalue annihilation, the shifting method clearly establishes its superiority. For the ARC2D code, an efficiency of 20 to 33% has been achieved by this method.