Spectral hp Element Methods for CFD

George Em Karniadakis, Spencer J. Sherwin0195102266, 9780195102260, 9781423741190

Traditionally spectral methods in fluid dynamics were used in direct and large eddy simulations of turbulent flow in simply connected computational domains. The methods are now being applied to more complex geometries, and the spectral/hp element method, which incorporates both multi-domain spectral methods and high-order finite element methods, has been particularly successful. This book provides a comprehensive introduction to these methods. Written by leaders in the field, the book begins with a full explanation of fundamental concepts and implementation issues. It then illustrates how these methods can be applied to advection-diffusion and to incompressible and compressible Navier-Stokes equations. Drawing on both published and unpublished material, the book is an important resource for experienced researchers and for those new to the field.

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Table of contents :
Contents……Page 8
1.1 The Basic Equations of Fluid Dynamics……Page 14
1.2 Numerical Discretizations……Page 19
2 Fundamental Concepts in One Dimension……Page 28
2.1 Method of Weighted Residuals……Page 29
2.2 Galerkin Formulation……Page 32
2.3 One-Dimensional Expansion Bases……Page 44
2.4 Numerical Integration……Page 67
2.5 Differentiation……Page 69
2.6 Convergence Examples……Page 72
3 Multidimensional Expansion Bases……Page 75
3.1 Expansions in Structured Domains……Page 76
3.2 Expansions in Unstructured Domains……Page 83
3.3 Expansions in Homogeneous Domains……Page 108
4.1 Local Elemental Operations……Page 109
4.2 Global Operations……Page 145
4.3 Boundary Representation……Page 165
5.1 The Need for Local Refinement……Page 179
5.2 Interface Conditions and Implementation……Page 181
5.3 Iterative Patching……Page 183
5.4 Constrained Approximation……Page 190
5.5 Mortar Patching……Page 192
6 Advection Equation……Page 200
6.1 Galerkin Discretization……Page 202
6.2 Temporal Discretization……Page 205
6.3 Eigen-Spectrum of the Galerkin Advection Operator……Page 208
6.4 Discontinuous Galerkin Discretization……Page 217
6.5 Convergence……Page 220
7.1 Galerkin Discretization……Page 224
7.2 Eigen-Spectrum of Laplacian Operator……Page 228
7.3 Convergence……Page 235
7.4 Non-Smooth Domains……Page 238
7.5 Mixed and Discontinuous Galerkin Discretization……Page 247
8.1 Variational Formulation……Page 251
8.2 Coupled Methods for Primitive Variables……Page 255
8.3 Splitting Methods for Primitive Variables……Page 260
8.4 Velocity-Vorticity Formulation……Page 273
8.5 The Gauge Method……Page 281
9.1 Exact Navier-Stokes Solutions……Page 282
9.2 Direct Numerical Simulations – DNS……Page 289
9.3 Large Eddy Simulations – LES……Page 303
9.4 Dynamic (dDNS) versus Static DNS……Page 311
10.1 Discontinuous Solutions and High Order……Page 317
10.2 Conservative Formulation……Page 319
10.3 Monotonicity……Page 330
10.4 Euler Equations……Page 339
10.5 Navier-Stokes Equations……Page 349
10.6 Shock-Fitting Techniques……Page 357
Appendices……Page 362
A. Jacobi Polynomials……Page 363
B. Gauss-Type Integration……Page 366
B.1 Jacobi Formulae……Page 367
B.2 Evaluation of the Zeros of Jacobi Polynomials……Page 369
C. Collocation Differentiation……Page 371
C.1 Jacobi Formulae……Page 372
D.1 Modal Basis……Page 375
D.2 Nodal Basis……Page 381
E.1 One dimension……Page 384
E.2 Two dimensions……Page 385
E.3 Three dimensions……Page 386
References……Page 388
D……Page 399
I……Page 400
P……Page 401
V……Page 402
Z……Page 403