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Author Gosse, Laurent.

Title Computing Qualitatively Correct Approximations of Balance Laws : Exponential-Fit, Well-Balanced and Asymptotic-Preserving
Published Dordrecht : Springer, 2013

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Description 1 online resource (346 pages)
Series SIMAI Springer Series ; v. 2
SIMAI Springer series ; 2.
Contents 880-01 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
880-01/(S 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
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
Notes 3.2.1 Wave-Front Tracking Approximations
Bibliography Includes bibliographical references and index
Notes Print version record
Subject Elasticity.
Conservation laws (Mathematics)
Elasticity
SCIENCE -- Nanoscience.
Conservation laws (Mathematics)
Elasticity
Form Electronic book
ISBN 9788847028920
8847028922