Stability and convergence analysis of implicit upwind schemes

Abstract Von Neumann stability analysis and convergence studies are used to parametrically study the performance capabilities of implicit upwind schemes, such as alternating direction implicit (ADI), diagonally dominant ADI (DDADI), point Gauss–Seidel (DDLU) and line Gauss–Seidel (DDLGS) methods. All the schemes are expressed in a common approximate factorization form, and multi-sweep strategies are formulated within a generalized dual-time framework. Stability analysis reveals that all the schemes are conditionally stable, although they may be rendered unconditionally stable by employing a sufficient number of inner sweeps. Study of high aspect ratio effects reveals the importance of using the min-CFL time-step definition and viscous preconditioning based upon the min-CFL and max-VNN. Stability and convergence studies reveal that the ADI and DDLGS schemes possess attractive convergence properties at all aspect ratios for 2D problems, while the DDADI and DDLU schemes perform worse at high aspect ratios.

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