# Aspects of the recursive projection method applied to flow calculations

Sammanfattning: In this thesis, we have investigated the Recursive Projection Method, RPM, as an accelerator for computations of both steady and unsteady flows, and as a stabilizer in a bifurcation analysis.The criterion of basis extraction is discussed. It can be interpreted as a tolerance for the accuracy of the eigenspace spanned by the identified basis, alternatively it can be viewed as a criterion when the approximative Krylov sequence becomes numerically rank deficient.Steady state calculations were performed on two different turbulent test-cases; a 2D supersonic nozzle flow with the Spalart-Allmaras 1-equation model and a 2D sub-sonic airfoil simulation using the κ - ε model. RPM accelerated the test-cases with a factor between 2 and 5.In multi-scale problems, it is often of interest to model the macro-scale behavior, still retaining the essential features of the full systems. The ``coarse time stepper'' is a heuristic approach for circumventing the analytical derivation of models. The system studied here is a linear lattice of non-linear reaction sites coupled by diffusion. After reformulation of the time-evolution equation as a fixed-point scheme, RPM coupled with arc-length continuation is used to calculate the bifurcation diagrams of the effective (but analytically unavailable) equation.Within the frame-work of dual time-stepping, a common approach in unsteady CFD-simulation, RPM is used to accelerate the convergence. Two test-cases were investigated; the von Karman vortex-street behind a cylinder at Re=100, and the periodic shock oscillation of a symmetric airfoil at M ∞ = 0.76 with a Reynolds number Re=11 x 106.It was believed that once a basis had been identified, it could be retained for several steps. The simulations usually showed that the basis could only be retained for one step.The need for updating the basis motivates the use of Krylov methods. The most common method is the (Block-) Arnoldi algorithm. As the iteration proceeds, Krylov methods become increasingly expensive and restart is required. Two different restart algorithm were tested. The first is that of Lehoucq and Maschhoff, which uses a shifted QR iteration, the second is a block extension of the single-vector Arnoldi method due to Stewart. A flexible hybrid algorithm is derived combining the best features of the two.

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