preconditioners for linearized discrete compressible euler equations

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Bernhard Pollul bering, the PBGS-preconditioned BiCGSTAB method does not converge in most cases. With WRG numbering, though, even with large CFL-numbers (e.g. 1000, 5000) the linear solver converges in most cases if we use PBGS. The WRG algorithm is black-box and significantly improves the robustness and efficiency of the point-block Gauss-Seidel preconditioner. For further explanation and properties of this and other renumbering algorithms we refer to [33]. 5 CONCLUSIONS We summarize the main conclusions. Already for our relatively simple model problems the PBSPAI(0) method turns out to be a poor preconditioner. This method should not be used in a Newton-Krylov method for solving compressible Euler equations. Both for model problems and a realistic application (QUADFLOW solver for NACA0012 airfoil) the efficiencies of the PBGS preconditioner and the PBILU(0) method are comparable. For our applications the PBILU(1) and PBILU(2) preconditioners are less efficient than the PBILU(0) preconditioner. To achieve a robust preconditioner PBILU(0) and PBGS can be coupled with a suitable ordering algorithm. Acknowledgement The experiments in section 4.3 are carried ot using the QUADFLOW solver developed in the Collaborative Research Center SFB 401 [37]. The authors acknowledge the fruitful collaboration with several members of the QUADFLOW research group. REFERENCES [1] K. Ajmani, W.-F. Ng and M. Liou, Preconditioned Conjugate Gradient Methods for the Navier-Stokes Equations, J. Comp. Phys. 110 68–81 (1994). [2] E. F. D’Azevedo, P.A. Forsyth and W.-P. Tang, Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems, SIAM J. Matrix Anal. Appl. 13(3) 944–961 (1992) [3] S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. Knepley, L. C. McInnes. B. F. Smith and H. Zhang, PETSc, http://www-fp.mcs.anl.gov/petsc/ (2001). [4] J. Ballmann (editor), Flow Modulation and Fluid-Structure-Interaction at Airplane Wings, Numerical Notes on Fluid Mechanics 84, Springer (2003). [5] S. T. Barnard and M. J. Grote, A Block Version of the SPAI Preconditioner, in Proc. 9th SIAM Conf. On Parallel Process. for Sci. Comp. (1999). [6] P. Batten, M. A. Leschziner and U.C. Goldberg, Average-State Jacobians and Implicit Methods for Compressible Viscours and Turbulent Flows, J. Comp. Phys. 137, 38–78 (1997). 16
Bernhard Pollul [7] Bey, J. Wittum, Downwind numbering: robust multigrid for convection-diffusion problems, App. Num. Math. 23 177–192 (1997). [8] K.H. Brakhage and S. Müller, Algebraic-hyperbolic Grid Generation with Precise Control of Intersection of Angles, Int. J. Numer. Meth. Fluids. 33, 89–123 (2000). [9] F. Bramkamp, B. Gottschlich-Müller, M. Hesse, Ph. Lamby, S. Müller, J. Ballmann, K.-H. Brakhage, W. Dahmen, H-adaptive Multiscale Schemes for the Compressible Navier-Stokes Equations: Polyhedral Discretization, Data Compression and Mesh Generation, in [4], 125–204 (2001). [10] F. Bramkamp, Ph. Lamby and S. Mller, An adaptive multiscale finite volume solver for unsteady and steady state flow comput., J. Comp. Physics, 197/2 460–490 (2004). [11] A. Brandt, Multi-level Adaptive Solutions to Boundary Value Problems, Math. Comp. 31, 333–390 (1977). [12] O. Bröker, M.J. Grote, C. Mayer, A. Reusken, Robust Parallel Smoothing for Multigrid via Sparse Approximate Inverses, SIAM J. Sci. Comput. 23 1396–1417 (2001). [13] E. Cuthill, Several strategies for reducing the band width of matrices. Sparse Matrices and Their Appl., D. J. Rose and R. A. Willoughby, eds., New York, 157–166 (1997). [14] E. Cuthill, J. McKee, Reducing the bandwidth of sparse symmetric matrices. in: Proc. ACM Nat. Conf., New York, 157–172 (1969). [15] M.J. Grote and T. Huckle, Parallel Preconditioning with Sparse Approximate Inverses, SIAM J. Scient. Comput. 18(3) (1997). [16] W. Hackbusch, Multi-grid Methods and Applications, Springer (1985). [17] W. Hackbusch, On the Feedback Vertex Set for a Planar Graph, Computing 58 129–155, (1997). [18] D. Hänel and F. Schwane, An Implicit Flux-Vector Splitting Scheme for the Computation of Viscous Hypersonic Flow, AIAA paper 0274 (1989). [19] C. Hirsch, Numerical computation of internal and external flows: computational methods for inviscid and viscous flows, 2, Wiley (1988). [20] A. Jameson, Solution of the Euler Equations for Two-Dimensional Transonic Flow by a Multigrid Method, Appl. Math. Comp. 13 327–356 (1983). [21] A. Jameson, D.A. Caughey, How Many Steps are Required to Solve the Euler Equations of Steady, Compressible Flow: in search of a Fast Solution Algorithm, AIAA paper 2673 (2001). 17
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