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E-raamat: Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues

Series edited by (Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany), Edited by (Universitat Zurich, Switzerland), Edited by (Brown University, RI, USA), Series edited by , Series edited by , Series edited by (Department of Applied Mathematics and Applied Physics, Columbia University, New York, NY USA)
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Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern IssuesSuurem pilt
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Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations.

  • Provides detailed, cutting-edge background explanations of existing algorithms and their analysis
  • Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications
  • Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage

Muu info

Cutting-edge explanations of existing algorithms and their analysis for readers working on the theoretical aspects of algorithm development and numerical analysis
Contributors xv
Editors' Introduction xvii
1 Cut Cells: Meshes and Solvers 1(22)
M. Berger
1 Introduction
1(2)
2 Brief Early History
3(2)
3 Mesh Generation
5(3)
4 Data Structures and Implementation Issues
8(2)
5 Finite Volume Methods for Cut Cells
10(8)
5.1 Steady-State Solution Techniques
12(1)
5.2 Explicit Time-dependent Solution Techniques
13(4)
5.3 Viscous Flows
17(1)
6 Conclusions
18(1)
Acknowledgements
18(1)
References
18(5)
2 Inverse Lax-Wendroff Procedure for Numerical Boundary Treatment of Hyperbolic Equations 23(30)
C.W. Shu
S. Tan
1 Introduction
24(3)
2 Problem Description and Interior Schemes
27(1)
3 Numerical Boundary Conditions for Static Geometry
28(10)
3.1 One-Dimensional Scalar Conservation Laws: Smooth Solutions
28(3)
3.2 One-Dimensional Scalar Conservation Laws: Solutions Containing Discontinuities
31(3)
3.3 Two-Dimensional Euler Equations in Static Geometry
34(4)
4 Moving Boundary Treatment for Compressible Inviscid Flows
38(4)
5 Numerical Results
42(6)
6 Conclusions and Future Work
48(1)
References
49(4)
3 Multidimensional Upwinding 53(28)
P. Roe
1 Introduction
53(4)
2 Why Multidimensional Methods?
57(4)
2.1 Dimensional Splitting and One-Dimensional Upwinding
58(3)
3 Oblique Wave Methods
61(1)
4 "Corner Transport" Methods
62(2)
5 Edges or Corners?
64(1)
6 When "Upwinding" Is Not Needed
65(2)
7 Bicharacteristic Methods
67(2)
8 Residual Distribution
69(6)
8.1 The N Scheme
70(1)
8.2 The NN Scheme
71(1)
8.3 Systems of Equations
72(1)
8.4 Unsteady Problems
72(1)
8.5 Wave Models
73(1)
8.6 Elliptic-Hyperbolic Splitting
73(2)
9 The Poisson Formulas
75(3)
9.1 Application to the Euler Equations
76(2)
10 Concluding Remarks
78(1)
References
78(3)
4 Bound-Preserving High-Order Schemes 81(22)
Z. Xu
X. Zhang
1 Introduction
82(1)
2 A Bound-Preserving Limiter for Approximation Polynomials
83(10)
2.1 First-Order Monotone Schemes
83(1)
2.2 The Weak Monotonicity in High-Order Finite Volume Schemes
84(2)
2.3 A Simple and Efficient Scaling Limiter
86(5)
2.4 SSP High-Order Time Discretizations
91(1)
2.5 Extensions and Applications
92(1)
3 Bound-Preserving Flux Limiters
93(4)
3.1 Basic Idea and Framework
93(2)
3.2 Decoupling for the Flux Limiting Parameters
95(2)
4 Concluding Remarks
97(1)
Acknowledgements
98(1)
References
98(5)
5 Asymptotic-Preservinu Schemes for Multiscale Hyperbolic and Kinetic Equations 103(28)
J. Hu
S. Jin
Q. Li
1 Introduction
104(1)
2 Basic Design Principles of AP Schemes-Two Illustrative Examples
105(5)
2.1 The Jin-Xin Relaxation Model
105(2)
2.2 The BGK Model
107(3)
3 AP Schemes for General Hyperbolic and Kinetic Equations
110(8)
3.1 AP Schemes Based on Penalization
111(4)
3.2 AP Schemes Based on Exponential Reformulation
115(2)
3.3 AP Schemes Based on Micro-Macro Decomposition
117(1)
4 Other Asymptotic Limits and AP Schemes
118(5)
4.1 Diffusion Limit of Linear Transport Equation
118(1)
4.2 High-Field Limit
119(1)
4.3 Quasi-Neutral Limit in Plasmas
120(1)
4.4 Low Mach Number Limit of Compressible Flows
121(1)
4.5 Stochastic AP Schemes
122(1)
5 Conclusion
123(1)
Acknowledgements
123(1)
References
124(7)
6 Well-Balanced Schemes and Path-Conservative Numerical Methods 131(46)
M.J. Castro
T. Morales de Luna
C. Pares
1 Introduction
132(6)
2 Path-Conservative Numerical Schemes
138(4)
3 Some Families of Path-Conservative Numerical Schemes
142(6)
3.1 Godunov Method
142(1)
3.2 Simple Riemann Solvers
142(2)
3.3 Roe Methods
144(1)
3.4 Functional Viscosity Matrix Methods
145(3)
3.5 Other Path-Conservative Methods
148(1)
4 High-Order Schemes Based on Reconstruction of States
148(3)
5 Well-Balanced Schemes
151(15)
5.1 Well-Balanced Property for SRSs
154(1)
5.2 Well-Balanced HLL Scheme
155(1)
5.3 Well-Balanced SRSs
156(1)
5.4 Roe Method
156(1)
5.5 Well-Balanced Functional Viscosity Matrix Methods
157(1)
5.6 Generalized Hydrostatic Reconstruction
158(2)
5.7 Well-Balanced Methods for a Subset of Stationary Solutions
160(1)
5.8 High-Order Well-Balanced Schemes
161(5)
6 Convergence and Choice of Paths
166(3)
Acknowledgements
169(1)
References
169(8)
7 A Practical Guide to Deterministic Particle Methods 177(26)
A. Chertock
1 Introduction
178(3)
2 Description of the Particle Method
181(9)
2.1 Particle Approximation of the Initial Data
182(1)
2.2 Time Evolution of Particles
183(3)
2.3 Particle Function Approximations
186(4)
3 Remeshing for Particle Distortion
190(4)
3.1 Particle Weights Redistribution
191(2)
3.2 Particle Merger-A Local Redistribution Technique
193(1)
4 Applications to Convection-Diffusion Equations
194(4)
4.1 Particle Methods for Convection-Diffusion Equations
195(3)
Acknowledgements
198(1)
References
198(5)
8 On the Behaviour of Upwind Schemes in the Low Mach Number Limit: A Review 203(30)
H. Guillard
B. Nkonga
1 Introduction
204(1)
2 The Multiple Low Mach Number Limits of the Compressible Euler Equations
205(16)
2.1 Incompressible Limit
205(2)
2.2 Acoustic Limit
207(1)
2.3 Acoustic-Incompressible Interactions
208(5)
2.4 Finite Volume Schemes
213(2)
2.5 The Diagnosis
215(2)
2.6 The Remedies
217(4)
3 Numerical Illustrations
221(7)
3.1 Order 1: Quadrangular Cartesian Grids
221(2)
3.2 Order 1: Vertex-Centred Triangular Meshes
223(1)
3.3 Order 1: Cell-Centred Triangular Meshes
224(2)
3.4 Order 2: Vertex-Centred Triangular Meshes
226(2)
4 Conclusion
228(1)
References
228(5)
9 Adjoint Error Estimation and Adaptivity for Hyperbolic Problems 233(30)
P. Houston
1 Introduction
234(1)
2 Error Representation for Linear Problems
235(8)
2.1 Abstract Framework
236(4)
2.2 Stabilized FEMs for the Linear Transport Equation
240(3)
3 A Posteriori Error Estimation
243(4)
4 Nonlinear Hyperbolic Conservation Laws
247(4)
5 Practical Implementation and Adaptive Mesh Refinement
251(3)
5.1 Numerical Approximation of the Dual Problem
251(1)
5.2 Adaptive Mesh Refinement
252(1)
5.3 Bibliographical Comments
253(1)
6 Applications
254(2)
6.1 Inviscid Flow Around a BAC3-11 Airfoil
254(1)
6.2 Criticality Problems
255(1)
6.3 Bifurcation Problems
255(1)
7 Concluding Remarks and Outlook
256(2)
Acknowledgements
258(1)
References
258(5)
10 Unstructured Mesh Generation and Adaptation 263(40)
A. Loseille
1 Introduction
264(2)
1.1 Outline
266(1)
2 An Introduction to Unstructured Mesh Generation
266(3)
2.1 Surface Mesh Generation
266(1)
2.2 Volume Mesh Generation
267(2)
3 Metric-Based Mesh Adaptation
269(11)
3.1 Metric Tensors in Mesh Adaptation
271(1)
3.2 Techniques for Enhancing Robustness and Performance
272(2)
3.3 Metric-Based Error Estimates
274(2)
3.4 Controlling the Interpolation Error
276(1)
3.5 Geometric Estimate for Surfaces
277(1)
3.6 Boundary Layers Metric
278(2)
4 Algorithms for Generating Anisotropic Meshes
280(3)
4.1 Insertion and Collapse
280(2)
4.2 Optimizations and Enhancement for Unsteady Simulations
282(1)
5 Adaptive Algorithm and Numerical Illustrations
283(13)
5.1 Adaptive Loop
284(1)
5.2 A Wing-Body Configuration
285(1)
5.3 Transonic Flow Around a M6 Wing
286(2)
5.4 Direct Sonic Boom Simulation
288(2)
5.5 Boundary Layer Shock Interaction
290(4)
5.6 Double Mach Reflection and Blast Prediction
294(2)
6 Conclusion
296(1)
References
297(6)
11 The Design of Steady State Schemes for Computational Aerodynamics 303(48)
F.D. Witherden
A. Jameson
D.W. Zingg
1 Introduction
304(1)
2 Equations of Gas Dynamics and Spatial Discretizations
305(3)
3 Time-Marching Methods
308(24)
3.1 Model Problem for Stability Analysis of Convection Dominated Problems
308(1)
3.2 Multistage Schemes for Steady State Problems
309(3)
3.3 Implicit Schemes for Steady State Problems
312(6)
3.4 Acceleration Methods
318(5)
3.5 Multigrid Methods
323(4)
3.6 RANS Equations
327(5)
4 Newton-Krylov Methods
332(12)
4.1 Background
332(3)
4.2 Methodology
335(9)
5 Conclusions
344(1)
References
345(6)
12 Some Failures of Riemann Solvers 351(10)
R. Abgrall
1 Introduction
351(2)
2 Real Gas Effects
353(3)
2.1 Mixture of Gases
353(2)
2.2 Nonlinear Equation of State
355(1)
3 Multidimensional Effects
356(2)
4 Accuracy Effects
358(2)
References
360(1)
13 Numerical Methods for the Nonlinear Shallow Water Equations 361(24)
Y. Xing
1 Overview
362(1)
2 Mathematical Model
363(1)
3 Numerical Methods
364(13)
3.1 Numerical Methods for the Homogeneous Equations
364(4)
3.2 Well-Balanced Methods
368(6)
3.3 Positivity-Preserving Methods
374(3)
4 Shallow Water-Related Models
377(3)
4.1 Shallow Water Flows Through Channels With Irregular Geometry
377(1)
4.2 Shallow Water Equations on the Sphere
378(1)
4.3 Two-Layer Shallow Water Equations
379(1)
5 Conclusion Remarks
380(1)
Acknowledgements
380(1)
References
380(5)
14 Maxwell and Magnetohydrodynamic Equations 385(18)
C.D. Munz
E. Sonnendrucker
1 Introduction
386(1)
2 Maxwell's Equations
386(10)
2.1 The Model
386(1)
2.2 Generalised Maxwell's Equations: Correcting the Fields
387(2)
2.3 Cartesian Grids: Finite Difference and Spectral Methods
389(1)
2.4 FV Schemes
390(3)
2.5 Discontinuous Galerkin Schemes
393(1)
2.6 Finite Element Methods
394(2)
3 Magnetohydrodynamics
396(2)
3.1 The Model
396(1)
3.2 Discretization
397(1)
4 Conclusion
398(1)
References
398(5)
15 Deterministic Solvers for Nonlinear Collisional Kinetic Flows: A Conservative Spectral Scheme for Boltzmann Type Flows 403(32)
I.M. Gamba
1 Introduction
404(6)
1.1 Kinetic Evolution Models
404(1)
1.2 Binary Collisional Models and Double Mixing Convolution Forms
405(3)
1.3 Classical Elastic Collisional Transport Theory: The Boltzmann Equation
408(1)
1.4 Deterministic Solvers for Integral Equations of Boltzmann Type
409(1)
2 The Landau and Boltzmann Operators Relation Through Their Double Mixing Convolutional Forms
410(5)
2.1 The Grazing Collision Limit
413(2)
3 A Conservative Spectral Method for the Collisional Form
415(11)
3.1 Choosing a Computational Cut-Off Domain DL
416(2)
3.2 Fourier Series, Projections and Extensions
418(1)
3.3 A Conservative Spectral Method for the Homogeneous Boltzmann Equation
419(2)
3.4 Conservation Method-An Extended Isoperimetric Problem
421(4)
3.5 Discrete in Time Conservation Method: Lagrange Multiplier Method
425(1)
4 Local Existence, Convergence and Regularity for the Semidiscrete Scheme
426(5)
4.1 Local Existence
427(1)
4.2 Uniform Propagation of Numerical Unconserved Moments
428(1)
4.3 Uniform L2k Integrability Propagation
429(1)
4.4 Uniform Semidiscrete Hk Sobolev Regularity Propagation
430(1)
5 Final Comments and Conclusions
431(1)
Acknowledgements
432(1)
References
432(3)
16 Numerical Methods for Hyperbolic Nets and Networks 435(30)
S. Canic
M.L. Delle Monache
B. Piccoli
J.M. Qiu
J. Tambaca
1 Introduction
436(1)
2 Examples of Nets and Networks
437(13)
2.1 Examples of Hyperbolic Nets
438(8)
2.2 Examples of Hyperbolic Networks
446(4)
3 Numerics for Nets and Networks
450(10)
3.1 Finite Volume Methods
451(1)
3.2 Discontinuous Galerkin Methods
451(9)
References
460(5)
17 Numerical Methods for Astrophysics 465(14)
C. Klingenberg
1 Introduction
466(1)
2 Astrophysical Scales for Astrophysical Phenomena
466(1)
2.1 Spatial Scales
466(1)
2.2 Density and Temporal Scales
466(1)
3 Equations Used in Astrophysical Modelling
467(2)
3.1 Source Terms
468(1)
3.2 Additional Force Terms
468(1)
3.3 Equation of State
469(1)
4 Numerical Methods
469(4)
4.1 Finite Difference Methods
469(1)
4.2 Finite Volume Methods
470(1)
4.3 Discontinuous Galerkin Method
471(1)
4.4 N-Body Method
472(1)
4.5 Grid-Free Method: Smoothed Particle Hydrodynamics
472(1)
5 High-Performance Computing
473(1)
6 Astrophysical Codes
473(2)
7 Conclusion
475(1)
Acknowledgement
475(1)
References
475(4)
18 Numerical Methods for Conservation Laws With Discontinuous Coefficients 479(28)
S. Mishra
1 Introduction
480(1)
1.1 Conservation Laws With Coefficients
481(1)
2 Motivating Examples
481(4)
2.1 Multiphase Flows in Porous Media
481(1)
2.2 Traffic Flow
482(1)
2.3 Other Examples of Scalar Conservation Laws With Discontinuous Flux
483(1)
2.4 Wave Propagation in Heterogeneous Media
484(1)
2.5 Systems of Conservation Laws With Singular Source Terms
484(1)
2.6 Flows as Perturbations of Discontinuous Steady States
485(1)
3 A Brief Review of Available Theoretical Results
485(6)
4 Numerical Schemes
491(5)
4.1 Aligned Schemes
491(2)
4.2 Staggered Schemes
493(1)
4.3 Higher-Order Schemes
494(1)
4.4 Extensions and Other Approaches
495(1)
5 Numerical Experiments
496(4)
5.1 Numerical Experiment 1
496(1)
5.2 Numerical Experiment 2
497(1)
5.3 Numerical Experiment 3
498(2)
6 Summary and Open Problems
500(2)
Acknowledgement
502(1)
References
502(5)
19 Uncertainty Quantification for Hyperbolic Systems of Conservation Laws 507(38)
R. Abgrall
S. Mishra
1 Introduction
508(3)
1.1 Numerical Methods
509(1)
1.2 Uncertainty Quantification
510(1)
2 Random Fields and Random Entropy Solutions
511(4)
2.1 Modelling of Random Inputs
512(2)
2.2 Random Entropy Solutions
514(1)
3 sG Method for UQ
515(5)
3.1 Generalized Polynomial Chaos
516(1)
3.2 Standard sG Method
517(1)
3.3 gPC Expansions in the Entropy Variables
518(2)
4 Stochastic Collocation Methods
520(4)
4.1 Standard Stochastic Collocation Method
520(1)
4.2 Stochastic Finite Volume Methods
521(3)
5 Monte Carlo and Multilevel Monte Carlo Methods
524(5)
5.1 Monte Carlo Method
524(2)
5.2 Multilevel Monte Carlo Finite Volume Method
526(3)
6 Numerical Experiments
529(7)
6.1 Compressible Euler Equations
529(1)
6.2 Uncertain Orszag-Tang Vortex
530(3)
6.3 UQ for the Lituya Bay Mega-Tsunami
533(1)
6.4 A Random Kelvin-Helmholtz Problem
534(2)
7 Measure-Valued and Statistical Solutions
536(2)
8 Conclusion and Perspectives
538(2)
Acknowledgements
540(1)
References
540(5)
20 Multiscale Methods for Wave Problems in Heterogeneous Media 545(32)
A. Abdulle
P. Henning
1 Introduction
546(3)
2 Numerical Methods for the Wave Equation in Heterogeneous Media Without Scale Separation
549(14)
2.1 Approach 1-Harmonic Coordinate Transformations
551(2)
2.2 Approach 2-MsFEM Using Limited Global Information
553(2)
2.3 Approach 3-Flux-Transfer Transformations
555(3)
2.4 Approach 4-Localized Orthogonal Decomposition
558(3)
2.5 The Case of General Initial Values: G-Convergence and Perturbation Arguments
561(2)
3 Numerical Methods for the Wave Equation in Heterogeneous Media With Scale Separation
563(11)
3.1 Effective Model and Numerical Homogenization Method for Short-Time Wave Propagation
564(5)
3.2 Effective Model and Numerical Homogenization Method for Long-Time Wave Propagation
569(5)
Acknowledgement
574(1)
References
574(3)
Index 577
Rémi Abgrall is a professor at Universität Zürich Professor Chi-Wang Shu is a professor at Brown University, RI, USA
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