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E-raamat: Numerical Approximation of Hyperbolic Systems of Conservation Laws

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  • Formaat: PDF+DRM
  • Sari: Applied Mathematical Sciences 118
  • Ilmumisaeg: 28-Aug-2021
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781071613443
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Numerical Approximation of Hyperbolic Systems of Conservation LawsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Applied Mathematical Sciences 118
  • Ilmumisaeg: 28-Aug-2021
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781071613443
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This monograph is devoted to the theory and approximation by finite volume methods of nonlinear hyperbolic systems of conservation laws in one or two space variables. It follows directly a previous publication on hyperbolic systems of conservation laws by the same authors. Since the earlier work concentrated on the mathematical theory of multidimensional scalar conservation laws, this book will focus on systems and the theoretical aspects which are needed in the applications, such as the solution of the Riemann problem and further insights into more sophisticated problems, with special attention to the system of gas dynamics. This new edition includes more examples such as MHD and shallow water, with an insight on multiphase flows. Additionally, the text includes source terms and well-balanced/asymptotic preserving schemes, introducing relaxation schemes and addressing problems related to resonance and discontinuous fluxes while adding details on the low Mach number situation.
I Introduction 1(54)
1 Definitions and Examples
1(5)
2 Fluid Systems in Eulerian and Lagrangian Frames
6(14)
3 Some Averaged Models: Shallow Water, Flow in a Duct, and Two-Phase Flow
20(7)
4 Weak Solutions of Systems of Conservation Laws
27(10)
4.1 Characteristics in the Scalar One-Dimensional Case
27(3)
4.2 Weak Solutions: The Rankine-Hugoniot Condition
30(5)
4.3 Example of Nonuniqueness of Weak Solutions
35(2)
5 Entropy Solution
37(15)
5.1 A Mathematical Notion of Entropy
37(7)
5.2 The Vanishing Viscosity Method
44(6)
5.3 Existence and Uniqueness of the Entropy Solution in the Scalar Case
50(2)
Notes
52(3)
II Nonlinear Hyperbolic Systems in One Space Dimension 55(86)
1 Linear Hyperbolic Systems with Constant Coefficients
55(3)
2 The Nonlinear Case, Definitions and Examples
58(22)
2.1 Change of Variables, Change of Frame
60(6)
2.2 The Gas Dynamics Equations
66(9)
2.3 Ideal MHD
75(5)
3 Simple Waves and Riemann Invariants
80(12)
3.1 Rarefaction Waves
80(4)
3.2 Riemann Invariants
84(8)
4 Shock Waves and Contact Discontinuities
92(11)
5 Characteristic Curves and Entropy Conditions
103(13)
5.1 Characteristic Curves
103(4)
5.2 The Lax Entropy Conditions
107(3)
5.3 Other Entropy Conditions
110(6)
6 Solution of the Riemann Problem
116(4)
7 Examples of Systems of Two Equations
120(17)
7.1 The Case of a Linear or a Linearly Degenerate System
120(2)
7.2 The Riemann Problem for the p-System
122(11)
7.3 The Riemann Problem for the Barotropic Euler System
133(4)
Notes
137(4)
III Gas Dynamics and Reacting Flows 141(74)
1 Preliminaries
141(12)
1.1 Properties of the Physical Entropy
141(8)
1.2 Ideal Gases
149(4)
2 Entropy Satisfying Shock Conditions
153(18)
3 Solution of the Riemann Problem
171(17)
4 Reacting Flows: The Chapman-Jouguet Theory
188(19)
5 Reacting Flows: The Z.N.D. Model for Detonations
207(5)
Notes
212(3)
IV Finite Volume Schemes for One-Dimensional Systems 215(210)
1 Generalities on Finite Volume Methods for Systems
215(21)
1.1 Extension of Scalar Schemes to Systems: Some Examples
221(9)
1.2 L2 Stability
230(2)
1.3 Dissipation and Dispersion
232(4)
2 Godunov's Method
236(14)
2.1 Godunov's Method for Systems
236(4)
2.2 The Gas Dynamics Equations in a Moving Frame
240(2)
2.3 Godunov's Method in Lagrangian Coordinates
242(3)
2.4 Godunov's Method in Eulerian Coordinates (Direct Method)
245(1)
2.5 Godunov's Method in Eulerian Coordinates (Lagrangian Step + Projection)
246(3)
2.6 Godunov's Method in a Moving Grid
249(1)
3 Godunov-Type Methods
250(33)
3.1 Approximate Riemann Solvers and Godunov-Type Methods
250(9)
3.2 Roe's Method and Variants
259(10)
3.3 The H.L.L. Method
269(5)
3.4 Osher's Scheme
274(9)
4 Roe-Type Methods for the Gas Dynamics System
283(37)
4.1 Roe's Method for the Gas Dynamics Equations: (I) The Ideal Gas Case
283(11)
4.2 Roe's Method for the Gas Dynamics Equations: (II) The "Real Gas" Case
294(5)
4.3 A Roe-Type Linearization Based on Shock Curve Decomposition
299(4)
4.4 Another Roe-Type Linearization Associated with a Path
303(6)
4.5 The Case of the Gas Dynamics System in Lagrangian Coordinates
309(11)
5 Flux Vector Splitting Methods
320(9)
5.1 General Formulation
320(2)
5.2 Application to the Gas Dynamics Equations: (I) Steger and Warming's Approach
322(4)
5.3 Application to the Gas Dynamics Equations: (II) Van Leer's Approach
326(3)
6 Van Leer's Second-Order Method
329(25)
6.1 Van Leer's Method for Systems
329(4)
6.2 Solution of the Generalized Riemann Problem
333(3)
6.3 The G.R.P. for the Gas Dynamics Equations in Lagrangian Coordinates
336(9)
6.4 Use of the G.R.P. in van Leer's Method
345(9)
7 Kinetic Schemes for the Euler Equations
354(40)
7.1 The Boltzmann Equation
354(9)
7.2 The B.G.K. Model
363(5)
7.3 The Kinetic Scheme
368(20)
7.4 Some Extensions of the Kinetic Approach
388(6)
8 Relaxation Schemes
394(26)
8.1 Introduction to Relaxation
394(5)
8.2 Model Examples
399(8)
8.3 A Relaxation Scheme for the Euler System
407(13)
Notes
420(5)
V The Case of Multidimensional Systems 425(156)
1 Generalities on Multidimensional Hyperbolic Systems
425(14)
1.1 Definitions
425(3)
1.2 Characteristics
428(5)
1.3 Simple Plane Waves
433(4)
1.4 Shock Waves
437(2)
2 The Gas Dynamics Equations in Two Space Dimensions
439(29)
2.1 Entropy and Entropy Variables
440(3)
2.2 Invariance of the Euler Equations
443(7)
2.3 Eigenvalues
450(5)
2.4 Characteristics
455(5)
2.5 Plane Wave Solutions: Self-Similar Solutions
460(8)
3 Multidimensional Finite Difference Schemes
468(19)
3.1 Direct Approach
468(12)
3.2 Dimensional Splitting
480(7)
4 Finite-Volume Methods
487(46)
4.1 Definition of the Finite-Volume Method
488(11)
4.2 General Results
499(18)
4.3 Usual Schemes
517(16)
5 Second-Order Finite-Volume Schemes
533(14)
5.1 MUSCL-Type Schemes
533(13)
5.2 Other Approaches
546(1)
6 An Introduction to All-Mach Schemes for the System of Gas Dynamics
547(31)
6.1 The Low Mach Limit of the System of Gas Dynamics
548(4)
6.2 Asymptotic Analysis of the Semi-Discrete Roe Scheme
552(9)
6.3 An All-Mach Semi-Discrete Roe Scheme
561(7)
6.4 Asymptotic Analysis of the Semi-Discrete HLL Scheme
568(6)
6.5 An All-Mach Semi-Discrete HLL Scheme
574(4)
Notes
578(3)
VI An Introduction to Boundary Conditions 581(46)
1 The Initial Boundary Value Problem in the Linear Case
581(18)
1.1 Scalar Advection Equations
582(5)
1.2 One-Dimensional Linear Systems. Linearization
587(3)
1.3 Multidimensional Linear Systems
590(9)
2 The Nonlinear Approach
599(7)
2.1 Nonlinear Equations
599(3)
2.2 Nonlinear Systems
602(4)
3 Gas Dynamics
606(4)
3.1 Fluid Boundary (Linearized Approach)
607(3)
3.2 Solid or Rigid Wall Boundary
610(1)
4 Absorbing Boundary Conditions
610(8)
5 Numerical Treatment
618(7)
5.1 Finite Difference Schemes
618(3)
5.2 Finite Volume Approach
621(4)
Notes
625(2)
VII Source Terms 627(122)
1 Introduction to Source Terms
627(16)
1.1 Some General Considerations for Systems with Source Terms
628(1)
1.2 Simple Examples of Source Terms in the Scalar Case
629(3)
1.3 Numerical Treatment of Source Terms
632(7)
1.4 Examples of Systems with Source Terms
639(4)
2 Systems with Geometric Source Terms
643(22)
2.1 Nonconservative Systems
644(6)
2.2 Stationary Waves and Resonance
650(6)
2.3 Case of a Nozzle with Discontinuous Section
656(6)
2.4 The Example of the Shallow Water System
662(3)
3 Specific Numerical Treatment of Source Terms
665(14)
3.1 Some Numerical Considerations for Flow in a Nozzle
665(2)
3.2 Preserving Equilibria, Well-Balanced Schemes
667(8)
3.3 Schemes for the Shallow Water System
675(4)
4 Simple Approximate Riemann Solvers
679(26)
4.1 Definition of Simple Approximate Riemann Solvers
679(3)
4.2 Well-Balanced Simple Schemes
682(3)
4.3 Simple Approximate Riemann Solvers in Lagrangian or Eulerian Coordinates
685(3)
4.4 The Example of the Gas Dynamics Equations with Gravity and Friction
688(9)
4.5 Link with Relaxation Schemes
697(8)
5 Stiff Source Terms, Asymptotic Preserving Numerical Schemes
705(26)
5.1 Introduction
705(2)
5.2 Some Simple Examples
707(4)
5.3 Derivation of an AP Scheme for the Linear Model
711(10)
5.4 Euler System with Gravity and Friction
721(10)
6 Interface Coupling
731(15)
6.1 Introduction to Interface Coupling
731(3)
6.2 The Interface Coupling Condition
734(10)
6.3 Numerical Coupling
744(2)
Notes
746(3)
References 749(82)
Index 831
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