Spectral hp element methods for CFD 2nd Edition by George Em Karniadakis, Spencer Sherwin ISBN 0199671362 9780199671366

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Product details: 

ISBN 10: 0199671362 

ISBN 13: 9780199671366

Author: George Em Karniadakis, Spencer Sherwin

Table of contents: 

1 Introduction
1.1 The basic equations of fluid dynamics
1.1.1 Incompressible flow
1.1.2 Reduced models
1.2 Numerical discretisations
1.2.1 The finite element method
1.2.2 Spectral discretisation
1.2.3 Why high-order accuracy in CFD?
1.2.4 Structured versus unstructured discretisation
1.2.5 What is hp convergence?
2 Fundamental concepts in one dimension
2.1 Method of weighted residuals
2.2 Galerkin formulation
2.2.1 Descriptive formulation
2.2.2 Two-domain linear finite element example
2.2.3 Mathematical formulation
2.2.4 Mathematical properties of the Galerkin approximation
2.2.5 Residual equation for the C[sup(0)] test and trial functions
2.3 One-dimensional expansion bases
2.3.1 Elemental decomposition: the h-type extension
2.3.2 Polynomial expansions: the p-type extension
2.3.3 Modal polynomial expansions
2.3.4 Nodal polynomial expansions
2.4 Elemental operations
2.4.1 Numerical integration
2.4.2 Differentiation
2.5 Error estimates
2.5.1 h-convergence of linear finite elements
2.5.2 L[sup(2)] error of the p-type interpolation in a single element
2.5.3 General error estimates for hp elements
2.6 Implementation of a one-dimensional spectral/hp element solver
2.6.1 Exercises
2.6.2 Convergence examples
3 Multi-dimensional expansion bases
3.1 Quadrilateral and hexahedral tensor product expansions
3.1.1 Standard tensor product extensions
3.1.2 Polynomial space of tensor product expansions
3.2 Generalised tensor product modal expansions
3.2.1 Coordinate systems
3.2.2 Orthogonal expansions
3.2.3 Modified C[sup(0)] expansions
3.3 Non-tensorial nodal expansions in a simplex
3.3.1 The Lagrange polynomial and the Lebesgue constant
3.3.2 Generalised Vandermonde matrix
3.3.3 Electrostatic points
3.3.4 Fekete points
3.4 Other useful tensor product extensions
3.4.1 Nodal elements in a prismatic region
3.4.2 Expansions in homogeneous domains
3.4.3 Cylindrical domains
3.5 Exercises: multi-dimensional elemental mass matrices
4 Multi-dimensional formulation
4.1 Local elemental operations
4.1.1 Integration within the standard region Ω[sub(st)]4.1.2 Differentiation in the standard region Ω[sub(st)]4.1.3 Operations within general-shaped elements
4.1.4 Discrete evaluation of the surface Jacobian
4.1.5 Elemental projections and transformations
4.1.6 Sum-factorisation/tensor product operations
4.2 Global operations
4.2.1 Global assembly and connectivity
4.2.2 Global matrix system
4.2.3 Static condensation/substructuring
4.2.4 Global boundary system numbering and ordering to enforce Dirichlet boundary conditions
4.3 Pre- and post-processing issues
4.3.1 Boundary condition discretisation
4.3.2 Elemental boundary transformation
4.3.3 Mesh generation for spectral/hp element discretisation
4.3.4 Global coarse meshing
4.3.5 High-order mesh generation
4.3.6 Particle tracking in spectral/hp element discretisations
4.4 Exercises: implementation of a two–dimensional spectral/hp element solver for a global project
5 Diffusion equation
5.1 Galerkin discretisation of the Helmholtz equation
5.2 Numerical examples
5.3 Temporal discretisation
5.3.1 Forward multi-step schemes
5.3.2 Backward multi-step schemes
5.4 Eigenspectra and iterative solution of weak Laplacian
5.4.1 Time-step restriction and maximum eigenvalue growth
5.4.2 Iterative solution and preconditioners
5.5 Non-smooth domains
5.5.1 Laplace equation in two-dimensional domains
5.5.2 Laplace equation in three-dimensional domains
5.5.3 Poisson equation
5.5.4 Helmholtz equation
5.5.5 Singular basis
5.5.6 Eigenpair representation: Steklov formulation
5.5.7 Singularities and Stokes flow
5.6 Exercises: implementation of a two-dimensional spectral/hp element solver for a Helmholtz proble
6 Advection and advection-diffusion
6.1 Linear advection equation
6.1.1 Dispersion and diffusion errors
6.2 Galerkin and discontinuous Galerkin discretisations
6.2.1 Galerkin discretisation of the linear advection equation
6.2.2 Discontinuous Galerkin method
6.3 Eigenspectrum of weak advection operator
6.3.1 Numerical evaluation of the time-step restriction
6.3.2 Eigenspectrum of the weak advection operator in Ω[sub(st)]6.4 Semi-Lagrangian formulation for advection–diffusion
6.4.1 Strong semi-Lagrangian method
6.4.2 Auxiliary semi-Lagrangian method
6.4.3 Convergence, efficiency, and stability of semi-Lagrangian schemes
6.5 Wiggles and high order: stabilisation techniques
6.5.1 Filters and relaxation
6.5.2 Spectral vanishing viscosity (SVV)
6.5.3 Over-integration of the viscous Burgers equation
6.5.4 Superconsistent collocation for advection–diffusion
7 Non–conforming elements
7.1 Interface conditions and implementation
7.2 Iterative patching
7.2.1 One-dimensional discretisation
7.2.2 Two-dimensional discretisation
7.2.3 Variational formulation
7.2.4 Interpretation of the relaxation procedure
7.3 Constrained approximation
7.4 Mortar patching
7.4.1 Projection and non-conforming spaces
7.4.2 The discrete second-order problem
7.4.3 Implementation
7.4.4 Condition number of the Laplacian
7.5 Discontinuous Galerkin method (DGM)
7.5.1 An inconsistent formulation
7.5.2 Local discontinuous Galerkin method (LDG)
7.5.3 The Baumann-Oden discontinuous Galerkin method
7.5.4 A unified formulation
7.5.5 Compactness of the stencil
7.5.6 Eigenspectrum
7.5.7 Convergence rate
7.5.8 Examples and comparisons
7.5.9 Stabilisation
7.5.10 Discontinuous Galerkin versus mixed formulation
7.5.11 Which DGM version to use?
8 Algorithms for incompressible flows
8.1 Variational formulation
8.2 Coupled methods for primitive variables
8.2.1 The Uzawa algorithm
8.2.2 Substructured Stokes system
8.3 Splitting methods for primitive variables
8.3.1 First-order schemes
8.3.2 High-order schemes
8.3.3 The inf–sup condition
8.3.4 Comparisons and recommendations
8.4 Velocity–vorticity formulation
8.4.1 Semi-discrete equations
8.4.2 Influence matrix implementation
8.4.3 Penalty method implementation
8.4.4 Spatial discretisation
8.5 Least-squares method
8.5.1 Formulation
8.5.2 Performance
8.6 The gauge method
8.7 Discretisation of nonlinear terms
8.7.1 Spatial discretisation
8.7.2 Temporal discretisation: semi-Lagrangian method
9 Incompressible flow simulations: verification and validation
9.1 Exact Navier–Stokes solutions
9.1.1 Moffatt eddies
9.1.2 Wannier flow
9.1.3 Kovasznay flow
9.1.4 Triangular duct flow
9.1.5 The Taylor vortex
9.2 BiGlobal stability analysis of complex flows
9.2.1 Formulation of the linearised eigenproblem
9.2.2 Iterative solution of the eigenproblem
9.2.3 Floquet analysis
9.2.4 Applications of BiGlobal stability
9.3 Direct numerical simulations—DNS
9.3.1 Under-resoulution and diagnostics
9.3.2 Stabilisation at high Reynolds number
9.4 Large-eddy simulations—LES
9.4.1 Governing equations and filters
9.4.2 Subgrid models
9.5 Dynamic (dDNS) versus static DNS
9.5.1 p-refinement and p-threads
9.5.2 The three-step Texas algorithm
9.5.3 Non-conforming spectral element refinement
10 Hyperbolic conservation laws
10.1 Conservative formulation
10.1.1 Cell-averaging procedure
10.1.2 Reconstruction procedure
10.1.3 Interfacial constraint
10.1.4 Non-oscillatory approximation
10.2 Monotonicity
10.2.1 Flux-corrected transport (FCT)
10.2.2 Local projection limiting
10.3 Euler equations
10.3.1 One-dimensional equations
10.3.2 Two-dimensional equations
10.3.3 Discontinuous Galerkin method
10.4 Shallow-water equations
10.4.1 Governing equations
10.4.2 Discontinuous Galerkin formulation
10.4.3 Examples
10.4.4 Boussinesq equations
10.5 Navier-Stokes equations
10.5.1 Mixed and discontinuous Galerkin formulations
10.5.2 Convergence and simulations
10.5.3 A penalty formulation
10.5.4 Moving domains
10.5.5 Stability and over-integration
10.6 Shock-fitting techniques
10.7 Magneto-hydrodynamics (MHD)
10.7.1 Governing equations
10.7.2 V . B = 0 constraint
10.7.3 A discontinuous Galerkin MHD solver
10.7.4 Convergence and simulations

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George Em Karniadakis,Spencer Sherwin,Spectral