1 Introduction 1

1.1 The Basic Equations of Fluid Dynamics 1

1.2 Numerical Discretizations 6

2 Fundamental Concepts in One Dimension 15

2.1 Method of Weighted Residuals 16

2.2 Galerkin Formulation 19

2.3 One-Dimensional Expansion Bases 31

2.4 Numerical Integration 54

2.5 Differentiation 56

2.6 Convergence Examples 59

3 Multidimensional Expansion Bases 62

3.1 Expansions in Structured Domains 63

3.2 Expansions in Unstructured Domains 70

3.3 Expansions in Homogeneous Domains 95

4 Multidimensional Formulation 96

4.1 Local Elemental Operations 96

4.2 Global Operations 132

4.3 Boundary Representation 152

5 Geometrically Non-Conforming Elements 166

5.1 The Need for Local Renement 166

5.2 Interface Conditions and Implementation 168

5.3 Iterative Patching 170

5.4 Constrained Approximation 177

5.5 Mortar Patching 179

6 Advection Equation 187

6.1 Galerkin Discretization 189

6.2 Temporal Discretization 192

6.3 Eigen-Spectrum of the Galerkin Advection Operator 195

6.4 Discontinuous Galerkin Discretization 204

6.5 Convergence 207

7 Helmholtz Equation 211

7.1 Galerkin Discretization 211

7.2 Eigen-Spectrum of Laplacian Operator 215

7.3 Convergence 222

7.4 Non-Smooth Domains 225

7.5 Mixed and Discontinuous Galerkin Discretization 234

8 Incompressible Flows 238

8.1 Variational Formulation 238

8.2 Coupled Methods for Primitive Variables 242

8.3 Splitting Methods for Primitive Variables 247

8.4 Velocity-Vorticity Formulation 260

8.5 The Gauge Method 268

9 Flow Simulations 269

9.1 Exact Navier-Stokes Solutions 269

9.2 Direct Numerical Simulations - DNS 276

9.3 Large Eddy Simulations - LES 290

9.4 Dynamic (dDNS) versus Static DNS 298

10 Compressible Flows 304

10.1 Discontinuous Solutions and High Order 304

10.2 Conservative Formulation 306

10.3 Monotonicity 317

10.4 Euler Equations 326

10.5 Navier-Stokes Equations 336

10.6 Shock-Fitting Techniques 344

Appendices 349

A. Jacobi Polynomials 350

B. Gauss-Type Integration 353

B.1 Jacobi Formulae 354

B.2 Evaluation of the Zeros of Jacobi Polynomials 356

C. Collocation Differentiation 358

C.1 Jacobi Formulae 359

D. Continuous Expansion Basis 362

D.1 Modal Basis 362

D.2 Nodal Basis 368

E. Characteristic Flux Decomposition 371

E.1 One dimension 371

E.2 Two dimensions 372

E.3 Three dimensions 373

References 375

Index 385