Computing Qualitatively Correct Approximations of Balance Laws Exponential-Fit, Well-Balanced and Asymptotic-Preserving

Format:

Book

Author/Creator:

Gosse, Laurent

Series:

SIMAI Springer series 2

SIMAI Springer Series v. 2

Language:

English

Subjects (All):

Elasticity.

Conservation laws (Mathematics).

Medical Subjects:

Elasticity.

Physical Description:

1 online resource

Place of Publication:

Dordrecht Springer 2013

Summary:

Substantial effort has been drawn for years onto the development of (possibly high-order) numerical techniques for the scalar homogeneous conservation law, an equation which is strongly dissipative in L1 thanks to shock wave formation. Such a dissipation property is generally lost when considering hyperbolic systems of conservation laws, or simply inhomogeneous scalar balance laws involving accretive or space-dependent source terms, because of complex wave interactions. An overall weaker dissipation can reveal intrinsic numerical weaknesses through specific nonlinear mechanisms: Hugoniot curve

Contents:

TitlePage; Copyright; Preface; Acknowledgements; Acronyms; Contents; Introduction and Chronological Perspective; 1.1 The Leap from Crank-Nicolson to Scharfetter-Gummel 1.1.1 Limitations for Gradients Computed with Finite Differences; 1.1 The Leap from Crank-Nicolson to Scharfetter-Gummel 1.1.1 Limitations for Gradients Computed with Finite Differences; 1.1.2 Numerical Gradients as Local First Integrals of the Motion; 1.1.2 Numerical Gradients as Local First Integrals of the Motion; 1v>; 1v>; 1v<; 1v<; 1.2 Modular Programming and Its Shortcomings; 1.2 Modular Programming and Its Shortcomings

Contents note continued: 12.2.3. Asymptotic-Preserving with Parabolic Scaling

12.3. Inclusion of an External Force by a Vlasov Term

12.3.1. Burschka-Titulaer's Eigenfunctions for Linear Potential

12.3.2. Scattering Matrix and Well-Balanced Scheme

12.4. Burgers/Fokker-Planck Modeling of Two-Phase Sprays

12.4.1. Theoretical Results for an Elementary Model

12.4.2. Overall Well-Balanced Numerical Simulation

12.4.3. Various Numerical Results References

13. Model for Scattering of Forward-Peaked Beams

13.1. Analysis of the Forward-Backward Inlet Problem

13.2. Derivation and Testing of the Well-Balanced Scheme

13.2.1. Scattering Matrix and Godunov Discretization

13.2.2. Constant Maxwellian Stabilization in a Box

13.2.3. Pencil Beam in an Inhomogeneous Environment References

14. Linearized BGK Model of Heat Transfer

14.1. Introduction

14.1.1. Short Review of the Boltzmann Equation

14.1.2. Simplified Models and Their Fluid Dynamic Approximation

14.1.3. Main Objectives of the Chapter

14.2. Elementary Solutions for the Linearized BGK Model

14.2.1. Cercignani's Decomposition of a Time-Dependent Problem

14.2.2. Elementary Solutions of the Heat Transfer System

14.2.3. Consistency with Navier-Stokes-Fourier Equations

14.3. Well-Balanced and Analytical Discrete-Ordinate Method

14.3.1. Gaussian Quadrature in the Velocity Variable and ADO

14.3.2. Complete Time-Dependent Scheme for Heat Transfer

14.4. Balancing Steady-States with Non-Zero Macroscopic Flux

14.4.1. Details on the Stationary Equation

14.4.2. Steady-States with Non-Zero Macroscopic Velocity

14.5. Numerical Results for Heat Transfer and Sound Wave

14.5.1. Boundary Conditions for Walls with Different Temperatures

14.5.2. Walls with Different Accommodation Coefficients: α1not = to α2

14.5.3. Sound Wave in Rarefied Gas

14.6. What Happens When the Knudsen Number Becomes Small

14.6.1. Small Knudsen Number in the Whole Domain

14.6.2. Computational Domain Containing Rarefied and Fluid Areas References

15. Balances in Two Dimensions: Kinetic Semiconductor Equations Again

15.1. Construction of a Well-Balanced N-Scheme

15.1.1. Original 2D N-Scheme on a Cartesian Mesh

15.1.2. Implementation of the Source Term by Jump Relations

15.2. Application to Vlasov-BGK Semi-Conductors Model

15.2.1. Exact Jump Relations and Derivation of the N-Scheme

15.2.2. Assessment of the WB N-scheme without Bias

15.2.3. N-Scheme with Moderate Bias: φ(x-1)=01/2

15.2.4. N-Scheme with Stronger Bias: φ(x=1)=-1 References

16. Conclusion: Outlook and Shortcomings

16.1. Shortcomings Inherent to Godunov-Type Schemes

16.2. How the Book Was Planned

16.3. Outlook and Future Research Directions References

A. Non-Conservative Products and Locally Lipschitzian Paths References

B. Tiny Step Toward Hypocoercivity Estimates for Well-Balanced Schemes on 2 x 2 Models

B.1. Simple Estimates on the Continuous Model

B.1.1. Macroscopic Formulation and Inequalities

B.1.2. Hints about the Proof of the Energy Estimates

B.2. Mimicking on the Numerical Scheme

B.2.1. Cheap Convexity Dissipation Estimate

B.2.2. Difficulties in Manipulating Macroscopic Quantities References

C. Preliminary Analysis of the Errors for Vlasov-BGK

C.1. Error Propagation on the Kinetic Density

C.2. Error Propagation on the 3 Moments References

1.2.1 Well-Balanced to Control Stiffness and Averaging Errors1.2.1 Well-Balanced to Control Stiffness and Averaging Errors; 1x.(; 1x.(; 1.2.2 Singular Perturbation Theory and Asymptotic-Preserving; 1.2.2 Singular Perturbation Theory and Asymptotic-Preserving; 1.3 Organization of the Book; 1.3 Organization of the Book; 1.3.1 Hyperbolic Systems of Balance Laws; 1.3.1 Hyperbolic Systems of Balance Laws; 1.3.2 Weakly Nonlinear Kinetic Equations; 1.3.2 Weakly Nonlinear Kinetic Equations; References; References; Part I; Lifting a Non-Resonant Scalar Balance Law

2.1 Generalities about Scalar Laws with Source Terms2.1 Generalities about Scalar Laws with Source Terms; 2.1.1 Method of Characteristics and Shocks; 2.1.1 Method of Characteristics and Shocks; 2.1.2 Entropy Solution and Kružkov Theory; 2.1.2 Entropy Solution and Kružkov Theory; 2.1.3 Initial-Boundary Value Problem and Large-Time Behavior; 2.1.3 Initial-Boundary Value Problem and Large-Time Behavior; 2.2 Localization Process of the Source Term on a Discrete Lattice; 2.2 Localization Process of the Source Term on a Discrete Lattice; 2.2.1 Nonconservative Lifting of an Inhomogeneous Equation

2.2.1 Nonconservative Lifting of an Inhomogeneous Equation1; 1; 2.2.2 The Measure Source Term Revealed by the Weaklimit; 2.2.2 The Measure Source Term Revealed by the Weaklimit; 2.2.3 A L1 Contraction Result "à la Kružkov"; 2.2.3 A L1 Contraction Result "à la Kružkov"; 2.3 Time-Exponential Error Estimate for the Godunov Scheme 2.3.1 Decay of Riemann Invariants and Temple Compactness; 2.3 Time-Exponential Error Estimate for the Godunov Scheme 2.3.1 Decay of Riemann Invariants and Temple Compactness; 2.3.2 Error Estimates for One-Dimensional Balance Laws

2.3.2 Error Estimates for One-Dimensional Balance Laws2.3.3 Application to the Scalar Well-Balanced Scheme; 2.3.3 Application to the Scalar Well-Balanced Scheme; Notes; Notes; References; References; Lyapunov Functional for Linear Error Estimates; 3.1 Preliminaries 3.1.1 A Puzzling Numerical Example; 3.1 Preliminaries 3.1.1 A Puzzling Numerical Example; 3.1.2 Lifting of the Balance Law: Temple System Reformulation; 3.1.2 Lifting of the Balance Law: Temple System Reformulation; 3.2 Error Estimate for Non-ResonantWave-Front Tracking; 3.2 Error Estimate for Non-ResonantWave-Front Tracking

Machine generated contents note: 1. Introduction and Chronological Perspective

1.1. Leap from Crank-Nicolson to Scharfetter-Gummel

1.1.1. Limitations for Gradients Computed with Finite Differences

1.1.2. Numerical Gradients as Local First Integrals of the Motion

1.2. Modular Programming and Its Shortcomings

1.2.1. Well-Balanced to Control Stiffness and Averaging Errors

1.2.2. Singular Perturbation Theory and Asymptotic-Preserving

1.3. Organization of the Book

1.3.1. Hyperbolic Systems of Balance Laws

1.3.2. Weakly Nonlinear Kinetic Equations References

2. Lifting a Non-Resonant Scalar Balance Law

2.1. Generalities about Scalar Laws with Source Terms

2.1.1. Method of Characteristics and Shocks

2.1.2. Entropy Solution and Krulkov Theory

2.1.3. Initial-Boundary Value Problem and Large-Time Behavior

2.2. Localization Process of the Source Term on a Discrete Lattice

2.2.1. Nonconservative Lifting of an Inhomogeneous Equation

2.2.2. Measure Source Term Revealed by the Weak-* limit

2.2.3. L1 Contraction Result "à la Kru[ž]kov"

2.3. Time-Exponential Error Estimate for the Godunov Scheme

2.3.1. Decay of Riemann Invariants and Temple Compactness

2.3.2. Error Estimates for One-Dimensional Balance Laws

2.3.3. Application to the Scalar Well-Balanced Scheme References

3. Lyapunov Functional for Linear Error Estimates

3.1. Preliminaries

3.1.1. Puzzling Numerical Example

3.1.2. Lifting of the Balance Law: Temple System Reformulation

3.2. Error Estimate for Non-Resonant Wave-Front Tracking

3.2.1. Wave-Front Tracking Approximations

3.2.2. Stability Estimates for Wave-Front Tracking Approximations

3.2.3. Limit δ->0 and Deviation from Kru[ž]kov's Entropy Solution

3.3. Error Estimate for the Non-Resonant Godunov Scheme

3.3.1. Design of a "Wave-Front Tracking/Godunov Scheme"

3.3.2. Control of the Functional's Jump at Each Averaging Step

3.3.3. Linear L1 Error Estimate and Comparison with Kuznetsov

3.3.4. Decoupling of the Time t and Grid Size Δx in (3.22)

3.4. More Transient Numerical Evidence

3.4.1. Inhomogeneous N-Wave

3.4.2. LeVeque-Yee's Effect for Riccati Source Term

3.4.3. Stationary Roll-Wave References

4. Early Well-Balanced Derivations for Various Systems

4.1. Huang-Liu's Piecewise-Steady Scheme for Nozzle Flows

4.1.1. Generalities and Quasi-One Dimensional Flows

4.1.2. Derivation of the Piecewise-Steady Scheme

4.1.3. Relation with Quasi-Steady Wave Propagation Algorithm

4.2. Sod's Random Choice Method for Diffusion Problems

4.2.1. Derivation of Sod's Algorithm

4.2.2. Relation with Scharfetter-Gummel's Procedure

4.3. Special Case: a Model of Atmosphere with Gravity

4.4. General Localization Process for the Source Term

4.4.1. Preliminary Versions of the Well-Balanced Scheme

4.4.2. Passing from the Scalar Case to General Systems

4.4.3. Flux-Splitting and Relation with Huang-Liu' s Scheme References

5. Viscosity Solutions and Large-Time Behavior for Non-Resonant Balance Laws

5.1. Small BV Existence, Uniqueness and L1 Stability Results

5.1.1. Structural Hypotheses on the n x n System

5.1.2. Definition of Small BV Viscosity Solutions

5.1.3. Stepping Stones for Existence and Stability Results

5.2. Weak and Strong Results for the Large-Time Behavior

5.2.1. Genuine Non-Linearity and Decay of Positive Waves

5.2.2. Non-Interacting Homogeneous Waves and Stationary Solutions References

6. Kinetic Scheme with Reflections and Linear Geometric Optics

6.1. Alternative Derivation of the Well-Balanced Kinetic Scheme

6.1.1. Consistency vs

Notes:

3.2.1 Wave-Front Tracking Approximations

Print version record

Includes bibliographical references and index

Other Format:

Print version Gosse, Laurent. Computing Qualitatively Correct Approximations of Balance Laws : Exponential-Fit, Well-Balanced and Asymptotic-Preserving

ISBN:

9788847028920

8847028922

OCLC:

857909916

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Restricted for use by site license