Riemann Solvers and Numerical Methods for Fluid Dynamics (eBook)

Preface to the First Edition 6
Preface to the Third Edition 10
Contents 13
1 The Equations of Fluid Dynamics 23
1.1 The Euler Equations 24
1.2 Thermodynamic Considerations 27
1.3 Viscous Stresses 37
1.4 Heat Conduction 39
1.5 Integral Form of the Equations 40
1.6 Submodels 47
2 Notions on Hyperbolic Partial Differential Equations 63
2.1 Quasi–Linear Equations: Basic Concepts 64
2.2 The Linear Advection Equation 69
2.3 Linear Hyperbolic Systems 72
2.4 Conservation Laws 83
3 Some Properties of the Euler Equations 109
3.1 The One–Dimensional Euler Equations 109
3.2 Multi–Dimensional Euler Equations 125
3.3 Conservative Versus Non–Conservative Formulations 134
4 The Riemann Problem for the Euler Equations 137
4.1 Solution Strategy 138
4.2 Equations for Pressure and Particle Velocity 141
4.3 Numerical Solution for Pressure 147
4.4 The Complete Solution 154
4.5 Sampling the Solution 158
4.6 The Riemann Problem in the Presence of Vacuum 161
4.7 The Riemann Problem for Covolume Gases 165
4.8 The Split Multi–Dimensional Case 171
4.9 FORTRAN Program for Exact Riemann Solver 173
5 Notions on Numerical Methods 185
5.1 Discretisation: Introductory Concepts 185
5.2 Selected Difference Schemes 190
5.3 Conservative Methods 196
5.4 Upwind Schemes for Linear Systems 209
5.5 Sample Numerical Results 216
5.6 FORTRAN Program for Godunov’s Method 218
6 The Method of Godunov for Non–linear Systems 235
6.1 Bases of Godunov’s Method 235
6.2 The Godunov Scheme 238
6.3 Godunov’s Method for the Euler Equations 240
6.4 Numerical Results and Discussion 247
7 Random Choice and Related Methods 258
7.1 Introduction 258
7.2 RCM on a Non–Staggered Grid 259
7.3 A Random Choice Scheme of the Lax–Friedrichs Type 265
7.4 The RCM on a Staggered Grid 268
7.5 Random Numbers 271
7.6 Numerical Results 276
7.7 Concluding Remarks 277
8 Flux Vector Splitting Methods 286
8.1 Introduction 286
8.2 The Flux Vector Splitting Approach 287
8.3 FVS for the Isothermal Equations 291
8.4 FVS Applied to the Euler Equations 294
8.5 Numerical Results 301
9 Approximate–State Riemann Solvers 313
9.1 Introduction 313
9.2 The Riemann Problem and the Godunov Flux 314
9.3 Primitive Variable Riemann Solvers (PVRS) 317
9.4 Approximations Based on the Exact Solver 321
9.5 Adaptive Riemann Solvers 324
9.6 Numerical Results 326
10 The HLL and HLLC Riemann Solvers 334
10.1 Introduction 334
10.2 The Riemann Problem 336
10.3 The HLL Approximate Riemann Solver 339
10.4 The HLLC Approximate Riemann Solver 341
10.5 Wave–Speed Estimates 346
10.6 Summary of HLLC Fluxes 350
10.7 Contact Waves and Passive Scalars 352
10.8 Numerical Results 353
10.9 Closing Remarks 355
11 The Riemann Solver of Roe 364
11.1 Bases of the Roe Approach 365
11.2 The Original Roe Method 370
11.3 The Roe–Pike Method 377
11.4 An Entropy Fix 385
11.5 Numerical Results and Discussion 391
11.6 Extensions 392
12 The Riemann Solver of Osher 396
12.1 Osher’s Scheme for a General System 397
12.2 Osher’s Flux for the Isothermal Equations 404
12.3 Osher’s Scheme for the Euler Equations 411
12.4 Numerical Results and Discussion 423
12.5 Extensions 425
13 High–Order and TVD Methods for Scalar Equations 431
13.1 Introduction 431
13.2 Basic Properties of Selected Schemes 433
13.3 WAF–Type High Order Schemes 438
13.4 MUSCL–Type High–Order Methods 444
13.5 Monotone Schemes and Accuracy 458
13.6 Total Variation Diminishing ( TVD) Methods 469
13.7 Flux Limiter Methods 474
13.8 Slope Limiter Methods 498
13.9 Extensions of TVD Methods 504
13.10 Numerical Results for Linear Advection 505
14 High–Order and TVD Schemes for Non–Linear Systems 511
14.1 Introduction 511
14.2 CFL and Boundary Conditions 513
14.3 Weighted Average Flux (waf) Schemes 514
14.4 The MUSCL–Hancock Scheme 522
14.5 Centred TVD Schemes 529
14.6 Primitive–Variable Schemes 533
14.7 Some Numerical Results 541
15 Splitting Schemes for PDEs with Source Terms 548
15.1 Introduction 548
15.2 Splitting for a Model Equation 550
15.3 Numerical Methods Based on Splitting 552
15.4 Remarks on ODE Solvers 554
15.5 Concluding Remarks 558
16 Methods for Multi–Dimensional PDEs 560
16.1 Introduction 560
16.2 Dimensional Splitting 561
16.3 Practical Implementation of Splitting Schemes in Three Dimensions 566
16.4 Unsplit Finite Volume Methods 572
16.5 A Muscl–Hancock Finite Volume Scheme 578
16.6 WAF–Type Finite Volume Schemes 580
16.7 Non–Cartesian Geometries 591
17 Multidimensional Test Problems 602
17.1 Explosions and Implosions 603
17.2 Shock Wave Reflection from a Wedge 608
18 FORCE Fluxes in Multiple Space Dimensions 614
18.1 Introduction 614
18.2 Review of FORCE in One Space Dimension 617
18.3 FORCE Schemes on Cartesian Meshes 622
18.4 Properties of the FORCE Schemes 627
18.5 FORCE Schemes on General Meshes 634
18.6 Sample Numerical Results 638
18.7 Concluding Remarks 638
19 The Generalized Riemann Problem 641
19.1 Introduction 641
19.2 Statement of the Problem 645
19.3 The Cauchy–Kowalewski Theorem 647
19.4 A Method of Solution 651
19.5 Examples 658
19.6 Other Solvers 667
19.7 Concluding Remarks 669
20 The ADER Approach 670
20.1 Introduction 670
20.2 The Finite Volume Method 672
20.3 Second–Order Scheme for a Model Equation 678
20.4 Schemes of Arbitrary Accuracy 682
20.5 Sample Numerical Results 684
20.6 Concluding Remarks 688
21 Concluding Remarks 693
21.1 Summary of Numerical Aspects 693
21.2 Potential Applications 695
21.3 Current Research Topics 699
21.4 The NUMERICA Library 700
References 701
Index 733