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Numerical Methods for the Unsteady Compressible Navier-Stokes Equations

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Numerical Methods for the Unsteady Compressible Navier-Stokes Equations Numerical Methods for the Unsteady Compressible Navier-Stokes Equations Philipp Birken Numerical Methods for the Unsteady Compressible Navier-Stokes Equations Dr. Philipp Birken Habilitationsschrift am Fachbereich Mathematik und Naturwissenschaften der Universit¨atKassel Gutachter: Prof. Dr. Andreas Meister Zweitgutachterin: Prof. Dr. Margot Gerritsen Drittgutachterin: Prof. Dr. Hester Bijl Probevorlesung: 31. 10. 2012 3 There are all kinds of interesting questions that come from a know- ledge of science, which only adds to the excitement and mystery and awe of a flower. It only adds. I don't un- derstand how it subtracts. Richard Feynman 6 Acknowledgements A habilitation thesis like this is not possible without help from many people. First of all, I'd like to mention the German Research Foundation (DFG), which has funded my research since 2006 as part of the collaborative research area SFB/TR TRR 30, project C2. This interdisciplinary project has provided me the necessary freedom, motivation and optimal funding to pursue the research documented in this book. It also allowed me to hire students to help me in my research and I'd like to thank Benjamin Badel, Maria Bauer, Veronika Diba, Marouen ben Said and Malin Trost, who did a lot of coding and testing and created a lot of figures. Furthermore, I thank the colleagues from the TRR 30 for a very successful research cooperation and in particular Kurt Steinhoff for starting the project. Then, there are my collaborators over the last years, in alphabetical order Hester Bijl, who thankfully also acted as reviewer, Jurjen Duintjer Tebbens, Gregor Gassner, Mark Haas, Volker Hannemann, Stefan Hartmann, Antony Jameson, Detlef Kuhl, Andreas Meis- ter, Claus-Dieter Munz, Karsten J. Quint, Mirek Tuma and Alexander van Zuijlen. Fur- thermore, thanks to Mark H. Carpenter and Gustaf S¨oderlindfor a lot of interesting dis- cussions. I'd also like to thank the University of Kassel and the Institute of Mathematics, as well as the colleagues there for a friendly atmosphere. In particular, there are the colleagues from the Institute of Mathematics in Kassel: Thomas Hennecke, Gunnar Matthies, Bet- tina Messerschmidt, Sigrun Ortleb who also did a huge amount of proofreading, Martin Steigemann, Philipp Zardo and special thanks to my long time office mate Stefan Kopecz. Furthermore, there are the colleagues at Stanford at the Institute of Mathematical and Computational Engineering and the Aero-Astro department, where I always encounter a friendly and inspiring atmosphere. In particular, I'd like to thank the late Gene Golub, Michael Saunders and Antony Jameson for many interesting discussions and getting things started. Furthermore, I'd like to thank Margot Gerritsen for being tremendously helpful and taking on the second review. My old friend Tarvo Thamm for proofreading. Not the last and certainly not the least, there is my former PhD advisor Andreas Meister and reviewer of this habilitation thesis, who has organized my funding and supported and enforced my scientific career whereever possible. Finally, there is my Margit who had to endure this whole process as much as me and who makes things worthwhile. 7 Contents 1 Introduction 13 1.1 The method of lines . 14 1.2 Hardware . 17 1.3 Notation . 18 1.4 Outline . 18 2 The Governing Equations 19 2.1 The Navier-Stokes Equations . 19 2.1.1 Basic form of conservation laws . 20 2.1.2 Conservation of mass . 20 2.1.3 Conservation of momentum . 21 2.1.4 Conservation of energy . 22 2.1.5 Equation of state . 22 2.2 Nondimensionalization . 23 2.3 Source terms . 25 2.4 Simplifications of the Navier-Stokes equations . 26 2.5 The Euler Equations . 26 2.6 Boundary and Initial Conditions . 27 2.7 Boundary layers . 29 2.8 Laminar and turbulent flows . 29 2.8.1 Turbulence models . 30 2.9 Analysis of viscous flow equations . 32 2.9.1 Analysis of the Euler equations . 32 2.9.2 Analysis of the Navier-Stokes equations . 33 3 The Space discretization 35 3.1 Structured and unstructured Grids . 36 3.2 Finite Volume Methods . 37 3.3 The Line Integrals and Numerical Flux Functions . 38 3.3.1 Discretization of the inviscid fluxes . 39 9 10 CONTENTS 3.3.2 Low Mach numbers . 43 3.3.3 Discretization of the viscous fluxes . 45 3.4 Convergence theory for finite volume methods . 47 3.4.1 Hyperbolic conservation laws . 47 3.4.2 Parabolic conservation laws . 48 3.5 Boundary Conditions . 49 3.5.1 Fixed wall . 49 3.5.2 Inflow and outflow boundaries . 50 3.5.3 Periodic boundaries . 51 3.6 Source Terms . 52 3.7 Finite volume methods of higher order . 52 3.7.1 Convergence theory for higher order finite volume schemes . 52 3.7.2 Reconstruction . 53 3.7.3 Modification at the Boundaries . 54 3.7.4 Limiters . 55 3.8 Discontinuous Galerkin methods . 56 3.8.1 Polymorphic modal-nodal scheme . 60 3.8.2 DG Spectral Element Method . 61 3.8.3 Discretization of the viscous fluxes . 63 3.9 Convergence theory for DG methods . 65 3.10 Spatial Adaptation . 65 4 Time Integration Schemes 67 4.1 Order of convergence and order of consistency . 68 4.2 Stability . 69 4.2.1 The linear test equation, A- and L-stability . 69 4.2.2 TVD stability and SSP methods . 70 4.2.3 The CFL condition, Von-Neumann stability analysis and related topics 71 4.3 Stiff problems . 74 4.4 Backward Differentiation formulas . 75 4.5 Runge-Kutta methods . 76 4.5.1 Explicit Runge-Kutta methods . 77 4.5.2 DIRK methods . 78 4.5.3 Additive Runge-Kutta methods . 82 4.6 Rosenbrock-type methods . 82 4.7 Adaptive time step size selection . 85 4.8 Operator Splittings . 87 4.9 Alternatives to the method of lines . 89 4.9.1 Local time stepping Predictor-Corrector-DG . 90 CONTENTS 11 5 Solving equation systems 93 5.1 The nonlinear systems . 93 5.2 The linear systems . 95 5.3 Rate of convergence and error . 96 5.4 Termination criterias . 97 5.5 Fixed Point methods . 98 5.5.1 Splitting Methods . 99 5.5.2 Fixed point methods for nonlinear equations . 101 5.6 Multigrid methods . 102 5.6.1 Multigrid for linear problems . 102 5.6.2 Full Approximation Schemes . 105 5.6.3 Steady state solvers . 106 5.6.4 Multi-p methods . 108 5.6.5 Dual Time stepping . 109 5.6.6 Optimizing Runge-Kutta smoothers for unsteady flow . 111 5.6.7 Optimizing the smoothing properties . 113 5.6.8 Optimizing the spectral radius . 116 5.6.9 Numerical results . 122 5.7 Newton's method . 124 5.7.1 Choice of initial guess . 129 5.7.2 Globally convergent Newton methods . 129 5.7.3 Computation and Storage of the Jacobian . 130 5.8 Krylov subspace methods . 131 5.8.1 GMRES and related methods . 132 5.8.2 BiCGSTAB . 134 5.9 Jacobian Free Newton-Krylov methods . 135 5.10 Comparison of GMRES and BiCGSTAB . 137 5.11 Comparison of variants of Newton's method . 138 6 Preconditioning linear systems 143 6.1 Preconditioning for JFNK schemes . 144 6.2 Specific preconditioners . 145 6.2.1 Block preconditioners . 145 6.2.2 Splitting-methods . 145 6.2.3 ROBO-SGS . 146 6.2.4 ILU preconditioning . 147 6.2.5 Multilevel preconditioners . 148 6.2.6 Nonlinear preconditioners . 149 6.2.7 Other preconditioners . 153 6.2.8 Comparison of preconditioners . 154 6.3 Preconditioning in parallel . 156 6.4 Sequences of linear systems . 156 12 CONTENTS 6.4.1 Freezing and Recomputing . 156 6.4.2 Triangular Preconditioner Updates . 156 6.4.3 Numerical results . 161.
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