Page 10 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Contents 1X

3.2 Numerical Dispersion and Numerical Diffusion 264

3.3 Lax, Lax—Wendroff and MacCormack Methods 266
Diffusion Equation 272
4.1 Forward-Time Scheme 272

4.2 Centered-Time Scheme 273
4.3 Backward-Time Scheme 274
4.4 Increasing the Scheme’s Accuracy 275
4.5 Numerical Example 275

Burgers Equation Without Shock 277
5.1 Lax Scheme 277
5.2 Leap-Frog Scheme 278

5.3 Lax—Wendroff Scheme 279
Hyperbolic Equations 280

6.1 Discretization of Hyperbolic Equations 280
6.2 Discretization in the Presence of a Shock 285
6.3 Method of Characteristics 291

Elliptic Equations 295
7A Iterative Methods 296
7.2 Direct Method 304

7.3 Transonic Flows 307
7A Stokes’ Problem 312
Compact Finite Differences 320

8.1 The Compact Finite Differences Method (CFDM) 320
8.2 Approximation of the Derivatives 321
8.3 Fourier Analysis of the Errors 326
8.4 Combined Compact Differences Schemes 329

8.5 Supercompact Difference Schemes 333
Coordinate Transformation 335

9.1 Coordinate Stretching 338
9.2 Boundary-Fitted Coordinate Systems 339
9.3 Adaptive Grids 344

FINITE ELEMENT AND BOUNDARY ELEMENT
METHODS 345

1 Finite Element Method (FEM) 345
1.1 Flow in the Presence of a Permeable Wall 349
1.2 PDE-Toolbox of MATLAB 354

Least-Squares Finite Element Method (LSFEM) 356
2.1 First Order Model Problem 356