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Numerical Methods for Unsteady Compressible Flow Problems [Kõva köide]

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Numerical Methods for Unsteady Compressible Flow ProblemsSuurem pilt
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"Numerical Methods for Unsteady Compressible Flow Problems is written to give both mathematicians and engineers an overview of the state of the art in the field, as well as of new developments. The focus is on methods for the compressible Navier-Stokes equations, the solutions of which can exhibit shocks, boundary layers and turbulence. The idea of the text is to explain the important ideas to the reader, while giving enough detail and pointers to literature to facilitate implementation of methods and application of concepts. The book covers high order methods in space, such as Discontinuous Galerkin methods, and high order methods in time, in particular implicit ones. A large part of the text is reserved to discuss iterative methods for the arising large nonlinear and linear equation systems. Ample space is given to both state-of-the-art multigrid and preconditioned Newton-Krylov schemes. Features Applications to aerospace, high-speed vehicles, heat transfer, and more besides Suitable as a textbook for graduate-level courses in CFD, or as a reference for practitioners in the field"--

Numerical Methods for Unsteady Compressible Flow Problems is written to give both mathematicians and engineers an overview of the state of the art in the field, as well as of new developments. The focus is on methods for the compressible Navier-Stokes equations, the solutions of which can exhibit shocks, boundary layers and turbulence. The idea of the text is to explain the important ideas to the reader, while giving enough detail and pointers to literature to facilitate implementation of methods and application of concepts.

The book covers high order methods in space, such as Discontinuous Galerkin methods, and high order methods in time, in particular implicit ones. A large part of the text is reserved to discuss iterative methods for the arising large nonlinear and linear equation systems. Ample space is given to both state-of-the-art multigrid and preconditioned Newton-Krylov schemes.

Features

  • Applications to aerospace, high-speed vehicles, heat transfer, and more besides

    • Suitable as a textbook for graduate-level courses in CFD, or as a reference for practitioners in the field

    • Preface and acknowledgments xiii
    1 Introduction 1(8)
    1.1 The method of lines
    		2(3)
    1.2 Hardware
    		5(1)
    1.3 Notation
    		6(1)
    1.4 Outline
    		6(3)
    2 The governing equations 9(20)
    2.1 The Navier-Stokes equations
    		9(5)
    2.1.1 Basic form of conservation laws
    		10(1)
    2.1.2 Conservation of mass
    		11(1)
    2.1.3 Conservation of momentum
    		11(1)
    2.1.4 Conservation of energy
    		12(1)
    2.1.5 Equation of state and Sutherland law
    		13(1)
    2.2 Nondimensionalization
    		14(2)
    2.3 Source terms
    		16(1)
    2.4 Simplifications of the Navier-Stokes equations
    		17(1)
    2.5 The Euler equations
    		18(1)
    2.5.1 Properties of the Euler equations
    		18(1)
    2.6 Solution theory
    		19(1)
    2.7 Boundary and initial conditions
    		20(2)
    2.7.1 Initial conditions
    		20(1)
    2.7.2 Fixed wall conditions
    		21(1)
    2.7.3 Moving walls
    		21(1)
    2.7.4 Periodic boundaries
    		21(1)
    2.7.5 Farfield boundaries
    		21(1)
    2.8 Boundary layers
    		22(1)
    2.9 Laminar and turbulent flows
    		22(7)
    2.9.1 Turbulence models
    		24(9)
    2.9.1.1 RANS equations
    		24(1)
    2.9.1.2 Large Eddy Simulation
    		25(2)
    2.9.1.3 Detached Eddy Simulation
    		27(2)
    3 The space discretization 29(36)
    3.1 Structured and unstructured grids
    		30(1)
    3.2 Finite volume methods
    		31(2)
    3.3 The line integrals and numerical flux functions
    		33(10)
    3.3.1 Discretization of the inviscid fluxes
    		34(5)
    3.3.1.1 HLLC
    		35(2)
    3.3.1.2 AUSMDV
    		37(2)
    3.3.2 Low Mach numbers
    		39(1)
    3.3.3 Discretization of the viscous fluxes
    		40(3)
    3.4 Convergence theory for finite volume methods
    		43(2)
    3.4.1 Hyperbolic conservation laws
    		43(1)
    3.4.2 Parabolic conservation laws
    		44(1)
    3.5 Source terms
    		45(1)
    3.6 Finite volume methods of higher order
    		45(4)
    3.6.1 Convergence theory for higher-order finite volume schemes
    		45(1)
    3.6.2 Linear reconstruction
    		46(1)
    3.6.3 Limiters
    		47(2)
    3.7 Discontinuous Galerkin methods
    		49(9)
    3.7.1 Polymorphic modal-nodal scheme
    		53(1)
    3.7.2 DG Spectral Element Method
    		54(2)
    3.7.3 Discretization of the viscous fluxes
    		56(2)
    3.8 Convergence theory for DG methods
    		58(1)
    3.9 Boundary conditions
    		58(5)
    3.9.1 Implementation
    		59(1)
    3.9.2 Stability and the SBP-SAT technique
    		59(1)
    3.9.3 Fixed wall
    		60(1)
    3.9.4 Inflow and outflow boundaries
    		61(1)
    3.9.5 Periodic boundaries
    		62(1)
    3.10 Spatial adaptation
    		63(2)
    4 Time integration schemes 65(30)
    4.1 Order of convergence and order of consistency
    		66(1)
    4.2 Stability
    		66(6)
    4.2.1 The linear test equation, A- and L-stability
    		67(1)
    4.2.2 TVD stability and SSP methods
    		68(1)
    4.2.3 The CFL condition, von Neumann stability analysis and related topics
    		69(3)
    4.3 Stiff problems
    		72(2)
    4.4 Backward differentiation formulas
    		74(1)
    4.5 Runge-Kutta methods
    		75(7)
    4.5.1 Explicit Runge-Kutta methods
    		77(1)
    4.5.2 Fully implicit RK methods
    		78(1)
    4.5.3 DIRK methods
    		78(3)
    4.5.4 Additive Runge-Kutta methods
    		81(1)
    4.6 Rosenbrock-type methods
    		82(4)
    4.6.1 Exponential integrators
    		85(1)
    4.7 Adaptive time step size selection
    		86(3)
    4.8 Operator splittings
    		89(2)
    4.9 Alternatives to the method of lines
    		91(3)
    4.9.1 Space-time methods
    		91(1)
    4.9.2 Local time stepping Predictor-Corrector-DG
    		91(3)
    4.10 Parallelization in time
    		94(1)
    5 Solving equation systems 95(50)
    5.1 The nonlinear systems
    		95(2)
    5.2 The linear systems
    		97(2)
    5.3 Rate of convergence and error
    		99(1)
    5.4 Termination criteria
    		99(2)
    5.5 Fixed point methods
    		101(4)
    5.5.1 Stationary linear methods
    		101(2)
    5.5.2 Nonlinear variants of stationary methods
    		103(2)
    5.6 Multigrid methods
    		105(22)
    5.6.1 Multigrid for linear problems
    		106(2)
    5.6.2 Full Approximation Schemes
    		108(2)
    5.6.3 Smoothers
    		110(5)
    5.6.3.1 Pseudo time iterations and dual time stepping
    		110(5)
    5.6.3.2 Alternatives
    		115(1)
    5.6.4 Residual averaging and smoothed aggregation
    		115(1)
    5.6.5 Multi-p methods
    		116(1)
    5.6.6 State of the art
    		117(1)
    5.6.7 Analysis and construction
    		118(9)
    5.6.7.1 Smoothing factors
    		118(1)
    5.6.7.2 Example
    		119(2)
    5.6.7.3 Optimizing the spectral radius
    		121(1)
    5.6.7.4 Example
    		121(2)
    5.6.7.5 Numerical results
    		123(2)
    5.6.7.6 Local Fourier analysis
    		125(1)
    5.6.7.7 Generalized locally Toeplitz sequences
    		126(1)
    5.7 Newton's method
    		127(7)
    5.7.1 Simplified Newton's method
    		129(1)
    5.7.2 Methods of Newton type
    		129(1)
    5.7.3 Inexact Newton methods
    		130(1)
    5.7.4 Choice of initial guess
    		131(1)
    5.7.5 Globally convergent Newton methods
    		132(1)
    5.7.6 Computation and storage of the Jacobian
    		133(1)
    5.8 Krylov subspace methods
    		134(4)
    5.8.1 GMRES and related methods
    		135(3)
    5.8.1.1 GCR
    		137(1)
    5.8.2 BiCGSTAB
    		138(1)
    5.9 Jacobian-free Newton-Krylov methods
    		138(2)
    5.10 Comparison of GMRES and BiCGSTAB
    		140(2)
    5.11 Comparison of variants of Newton's method
    		142(3)
    6 Preconditioning linear systems 145(18)
    6.1 Preconditioning for JFNK schemes
    		146(1)
    6.2 Specific preconditioners
    		147(8)
    6.2.1 Block preconditioners
    		147(1)
    6.2.2 Stationary linear methods
    		147(1)
    6.2.3 ROBO-SGS
    		148(1)
    6.2.4 ILU preconditioning
    		149(1)
    6.2.5 Multilevel preconditioners
    		150(1)
    6.2.6 Nonlinear preconditioners
    		151(1)
    6.2.7 Other preconditioners
    		152(1)
    6.2.8 Comparison of preconditioners
    		153(2)
    6.3 Preconditioning in parallel
    		155(1)
    6.4 Sequences of linear systems
    		156(4)
    6.4.1 Freezing and recomputing
    		156(1)
    6.4.2 Triangular preconditioner updates
    		156(3)
    6.4.3 Numerical results
    		159(1)
    6.5 Discretization for the preconditioner
    		160(3)
    7 The final schemes 163(14)
    7.1 DIRK scheme
    		164(2)
    7.2 Rosenbrock scheme
    		166(2)
    7.3 Parallelization
    		168(1)
    7.4 Efficiency of finite volume schemes
    		168(3)
    7.5 Efficiency of Discontinuous Galerkin schemes
    		171(6)
    7.5.1 Polymorphic modal-nodal DG
    		171(3)
    7.5.2 DG-SEM
    		174(3)
    8 Thermal Fluid Structure Interaction 177(12)
    8.1 Gas quenching
    		177(1)
    8.2 The mathematical model
    		178(2)
    8.3 Space discretization
    		180(1)
    8.4 Coupled time integration
    		180(1)
    8.5 Dirichlet-Neumann iteration
    		181(2)
    8.5.1 Extrapolation from time integration
    		183(1)
    8.6 Alternative solvers
    		183(1)
    8.7 Numerical Results
    		184(5)
    8.7.1 Cooling of a flanged shaft
    		184(5)
    A Test problems 189(6)
    A.1 Shu-vortex
    		189(1)
    A.2 Supersonic flow around a cylinder
    		189(1)
    A.3 Wind turbine
    		190(1)
    A.4 Vortex shedding behind a sphere
    		191(4)
    B Coefficients of time integration methods 195(6)
    Bibliography 201(24)
    Index 225
    Philipp Birken is an associate professor for numerical analysis and scientific computing at the Centre for the Mathematical Sciences at Lund University, Sweden. He received his PhD in mathematics in 2005 from the University of Kassel, Germany, where he also received his habilitation in 2012. He was a PostDoc and a consulting assistant professor at the Institute for Computational & Mathematical Engineering at Stanford University, USA. His work is in numerical methods for compressible CFD and Fluid-Structure-Interaction.