Theory, Numerics and Applications of Hyperbolic Problems II: Aachen, Germany, August 2016 (Springer Proceedings in Mathematics & Statistics (237))

Contents
Proceedings of the 16th International Conference on Hyperbolic Problems: Theory, Numerics and Application
	Organizer’s Introduction
	Reference
Speakers at the 16th International Conference on Hyperbolic Problems that Contributed to these Proceedings
A Stochastic Galerkin Method  for the Fokker–Planck–Landau Equation with Random Uncertainties
	1 Introduction
	2 The gPC-sG Method for the FPL Equation  with Uncertainties
	3 The Spatial and Time Discretization
	4 Consistency Analysis of the gPC-sG Method  for the Collision Operator
	5 A Remark on High-Dimensional Random Spaces
	6 Numerical Results
		6.1 Random Initial Data: A Shock Tube Problem
		6.2 The Landau Damping
		6.3 A Random Neutralizing Background
		6.4 An Example with a Two-Dimensional Random Variable
		6.5 An Example with a Six-Dimensional Random Domain
	7 Conclusion
	References
On Robust and Adaptive Finite Volume Methods for Steady Euler Equations
	1 Introduction
	2 Finite Volume Framework for Steady Euler Equations
		2.1 Governing Equations
		2.2 Newton Linearization
	3 Solution Reconstruction
	4 Towards Efficiency
	5 Numerical Tests
		5.1 Subsonic Flow Through a Circular Cylinder
		5.2 Inviscid Flow Through a Channel with a Smooth Bump
		5.3 Transonic Flow Around a NACA 0012 Airfoil
	6 Conclusion
	References
The Burgers–Hilbert Equation
	1 The Burgers–Hilbert Equation
	2 Vorticity Discontinuities
	3 Smooth Solutions and Singularity Formation
	4 Shocks
	5 Small-Amplitude BH Dynamics
	6 Enhanced Life Span of Smooth Solutions
	7 Asymptotic Equation for the BH Equation
	References
General Linear Methods  for Time-Dependent PDEs
	1 Introduction
	2 Numerical Method
		2.1 The Hybridized Discontinuous Galerkin Method
		2.2 General Linear Methods
		2.3 Applying DIMSIMs to the HDG Method
	3 Numerical Results
		3.1 Linear Convection–Diffusion Equation
		3.2 Navier–Stokes Equations
	4 Conclusion and Outlook
	References
An Invariant-Region-Preserving (IRP) Limiter to DG Methods for Compressible Euler Equations
	1 Introduction
	2 The Limiter
		2.1 Averaging is Contraction
		2.2 Reconstruction
		2.3 Algorithm
	3 Numerical Tests
	4 Conclusion and Future Work
	References
β-Schemes with Source Terms  and the Convergence Analysis
	1 Introduction
	2 The Convergence of β-Schemes with Source Terms
	References
Existence of Undercompressive Shock Wave Solutions to the Euler Equations
	1 Introduction
	2 The Mathematical Model and the Main Result
		2.1 The Initial Boundary Value Problem
	3 Proof of the Main Result
	References
Some Numerical Results of Regional Boundary Controllability with Output Constraints
	1 Introduction
	2 Preliminaries and Notations
	3 Lagrangian Approach
	4 Numerical Approach and Simulations
	5 Conclusion
	References
Water Hammer Modeling for Water Networks via Hyperbolic PDEs  and Switched DAEs
	1 Introduction
	2 Mathematical Model
		2.1 Models of Water Flow in Pipe
		2.2 Other Network Elements
	3 Analysis of a Simple Water Network
		3.1 PDE Mode1
		3.2 Switched DAE Framework
	4 Discussion on Switched DAEs
	5 Comparison of both Modeling Approaches
	6 Conclusion
	References
Stability Criteria for Some System  of Delay Differential Equations
	1 Introduction
	2 Stability Criteria
	References
Bound-Preserving Reconstruction  of Tensor Quantities for Remap in ALE Fluid Dynamics
	1 Introduction
	2 Governing Equations—the Lagrangian Step
	3 Remapping of the Deviatoric Stress Tensor
		3.1 Component-Wise Remap and Limiting of Tensor S
		3.2 VIP Limiter for Tensors
		3.3 J2 Invariant Scaling Limiter
		3.4 J2 Invariant-Based Scalar Slope Limiter
		3.5 Independent Remap of S and J2
	4 Numerical Results—Cyclic Remapping of a Nonlinear Radial Distribution of the Deviatoric Stress Tensor
	5 Conclusion
	References
On Computing Compressible Euler Equations with Gravity
	1 Introduction
	2 Relaxation
	3 Well-Balanced Property
	4 Numerical Scheme
	5 Numerical Results
	References
On Well-Posedness for a Multi-particle Fluid Model
	1 Introduction
	2 Definition of Entropy Solutions
	3 Existence
	4 Uniqueness of Entropy Solutions
	References
On Quantifying Uncertainties for the Linearized BGK Kinetic Equation
	1 Introduction
	2 Setup
	3 Variation in u
	4 Variation in T
	References
Kinetic ES-BGK Models  for a Multi-component Gas Mixture
	1 The BGK Approximation
	2 Extensions to an ES-BGK Approximation
		2.1 Extension of the Single Relaxation Terms
		2.2 Alternative Extensions to an ES-BGK Model
	References
An Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method  for Conservation Laws: Entropy Stability
	1 Introduction
	2 The ALE-DG Method
		2.1 The Semi-discrete ALE-DG Discretization
		2.2 Cell Entropy Inequalities
	3 Conclusions
	References
Simplified Hyperbolic Moment Equations
	1 Introduction
	2 Moment Method for the BGK Equation
		2.1 Existing Hyperbolic Moment Models
	3 A New Simplified Hyperbolic Moment Model SHME
		3.1 Discussion of the SHME Model
		3.2 Relation to Discrete Velocity Model
	4 Simulation Results
	5 Conclusion
	References
Weakly Coupled Systems  of Conservation Laws on Moving Surfaces
	1 Introduction
	2 The Parabolic System
	3 Existence of an Entropy Solution
	References
A Phase-Field Model for Flows with Phase Transition
	1 Introduction
	2 A Phase-Field Model
	3 Non-conservative Mixed Form
		3.1 Discretization
	4 Numerical Examples
		4.1 Convergence Test
		4.2 Bubble Ensemble
	References
Mathematical Theory of Two-Phase Geochemical Flow with Chemical Species
	1 Introduction
	2 Eigenvalues, Eigenvectors and Bifurcations
		2.1 Bifurcation Structures in the Model
	3 Rankine–Hugoniot Locus
	4 Conclusions
	References
Localization of Adiabatic Deformations in Thermoviscoplastic Materials
	1 Introduction
	2 Main Results
		2.1 Self-similar Structure
		2.2 Main Theorem
		2.3 Emergence of Localization
	3 Existence via Geometric Singular Perturbation Theory
		3.1 Critical Manifold
		3.2 Chapman–Enskog-Type Reduction
		3.3 Confinement of the Orbit
	References
The Global Nonlinear Stability  of Minkowski Spacetime for Self-gravitating Massive Fields
	1 Introduction
	2 Self-gravitating Massive Fields
	3 The Wave-Klein-Gordon Formulation
	4 The Global Nonlinear Stability
	References
A Particle-Based Multiscale Solver  for Compressible Liquid–Vapor Flow
	1 Introduction
	2 The Macroscale Model: Compressible, Isothermal Euler Equations
	3 The Microscale Model: Particle Chain Model
		3.1 Micro-/Macroscale Conversion: Irving–Kirkwood Formulas
		3.2 The Microscopic Riemann Problem
		3.3 Discretization of the Particle System
	4 The Multiscale Model
		4.1 Model Reduction Algorithm
		4.2 Numerical Discretization of the Multiscale Model
	5 Numerical Simulations
	6 Conclusions
	References
Lp-Lq Decay Estimates for Dissipative Linear Hyperbolic Systems in 1D
	1 Introduction
	2 Proofs of Main Results
		2.1 Spectral Analysis
		2.2 Fundamental Solution
		2.3 Multiplier Estimates
		2.4 Proofs of Theorems 1 and 2
	References
A Numerical Approach of Friedrichs' Systems Under Constraints in Bounded Domains
	1 Introduction
	2 Description of the Scheme
	3 Previous Results on Constrained Friedrichs' Systems  in the Whole Space
	4 Stability in Time of Schemes
	5 The Simplified Model of the Dynamical Perfect Plasticity
	6 A Numerical Test on the Simplified Model  of the Dynamical Perfect Plasticity
	References
Lagrangian Representation for Systems of Conservation Laws: An Overview
	1 Introduction
	2 Analysis of BV Solutions to Scalar Conservation Laws
	3 Analysis of Linear Systems of Conservation Laws
	4 The Riemann Problem
	5 Definition of Lagrangian Representation for Systems
	6 Construction of a Lagrangian Representation
	References
Kinematical Conservation Laws  in Inhomogeneous Media
	1 Introduction
	2 Ray Coordinate System and Kinematical Conservation Laws
	3 Weakly Nonlinear Ray Theory (WNLRT)
		3.1 Transport Equation in the Ray Coordinate System
	4 Shock Ray Theory (SRT)
	5 Numerical Results
	References
Artificial Viscosity for Correction Procedure via Reconstruction Using Summation-by-Parts Operators
	1 Introduction
	2 CPR Methods Using SBP Operators
	3 Artificial Dissipation/Spectral Viscosity
		3.1 Continuous Setting
		3.2 Semidscrete setting
		3.3 Discrete setting
	4 Numerical results
	5 Conclusion and outlook
	References
On a Relation Between Shock Profiles and Stabilization Mechanisms  in a Radiating Gas Model
	1 Introduction
	2 Subcritical case (Proof of Theorem 1)
	3 Critical Case (Proof of Theorem 2)
	4 Supercritical Case (Proof of Theorem 5)
		4.1 Energy Estimates Away from the Discontinuity
		4.2 Energy Estimates over the Entire Domain
	References
On the Longtime Behavior of Almost Periodic Entropy Solutions to Scalar Conservation Laws
	1 Introduction
	2 Proof of Theorem 2
	3 Proof of Theorem 4
	References
Structure Preserving Schemes  for Mean-Field Equations of Collective Behavior
	1 Introduction
	2 Derivation of the Schemes
		2.1 One-dimensional Case
		2.2 The Multidimensional Case
	3 Main Properties
		3.1 Nonnegativity
		3.2 Entropy Property
	4 Numerics
	References
A Numerical Model for Three-Phase Liquid–Vapor–Gas Flows with Relaxation Processes
	1 Introduction
	2 Single-Velocity Multiphase Compressible Flow Model
		2.1 Hierarchy of Multiphase Relaxed Models and Speed  of Sound
	3 Numerical Method
		3.1 Solution of the Homogeneous System
		3.2 Relaxation Steps
	4 Numerical Experiments
		4.1 Three-Phase Water Cavitation Tube
		4.2 Underwater Explosion Close to a Rigid Surface
	5 Conclusions
	References
Feedback Stabilization of a Linear Fluid–Membrane System with Time Delay
	1 Introduction
	2 Generalized Traces for Some Graph Spaces
	3 Abstract Formulation and Well-Posedness of the System
	4 Spectral Properties
	5 Uniform Exponential Stability
	References
A Unified Hyperbolic Formulation  for Viscous Fluids and Elastoplastic Solids
	1 Introduction
	2 Physical Model
	3 Mathematical Model
	4 Numerical Results
		4.1 Non-equilibrium Sound Wave Propagation in a Viscous Gas
		4.2 Viscous Fluids and Elastoplastic Solids
	References
On the Transverse Diffusion of Beams  of Photons in Radiation Therapy
	1 Introduction
	2 Photon Transport Models
		2.1 A Kinetic Model
		2.2 The M1 Model
	3 Numerical Approach
		3.1 Relaxation Method
		3.2 A Numerical Scheme for 2D Equations
	4 A Numerical Experiment
	5 A Correction to Accurately Model Transverse Diffusion
		5.1 Bounds on the Eigenvalues of the Jacobian of the Flux
		5.2 The Modified Relaxation Parameters
		5.3 The New Numerical Scheme
		5.4 Results with the Modified Scheme
	6 Conclusion
	References
Numerical Viscosity in Large Time Step HLL-Type Schemes
	1 Introduction
	2 Preliminaries
		2.1 Problem Outline
		2.2 Numerical Methods
		2.3 Estimates for Wave Velocities  SL  and  SR
	3 Entropy Violation
		3.1 Modified Equation Analysis
	4 Results
		4.1 Sod Shock Tube
		4.2 Double Rarefaction Problem
		4.3 Woodward–Colella Blast-Wave Problem
	5 Conclusions
	References
Correction Procedure via Reconstruction Using Summation-by-Parts Operators
	1 Introduction
	2 Correction Procedure via Reconstruction
	3 Summation-by-Parts Operators
	4 Correction Procedure via Reconstruction Using Summation-by-Parts Operators
	5 Linear Stability
	6 Nonlinear Stability for Burgers' Equation
	7 Further Research
	References
A Third-Order Entropy Stable Scheme for the Compressible Euler Equations
	1 Introduction
	2 Mesh and Finite Difference Scheme
	3 SP-WENO
	4 Numerical Results with SP-WENO
	5 SP-WENOc: A Fix for Systems of Conservation Laws
	6 Numerical Results with SP-WENOc
	7 Conclusions
	References
Did Numerical Methods for Hyperbolic Problems Take a Wrong Turning?
	1 Introduction
	2 Godunov's Question
	3 Limitations of One-Dimensional Treatments
		3.1 Does Discontinuous Reconstruction Help?
	4 A Multidimensional Method
		4.1 Objectives and Tools
		4.2 Techniques
	5 The Full Euler Equations
	6 Summary
	References
Astrophysical Fluid Dynamics and Applications to Stellar Modeling
	1 Introduction
	2 The Challenges of Astrophysical Fluid Dynamics
	3 Application to Stars
		3.1 Why Stars?
		3.2 Stellar Fluid Dynamics
	4 Toward a New Generation of Stellar Models
		4.1 Example: Type Ia Supernova Simulations
		4.2 Example: Multidimensional Models of Stellar Interiors
		4.3 Example: Common Envelope Phases in Binary Stellar Evolution
	5 Conclusions
	References
Nonlinear Stability of Localized and Non-localized Vortices in Rotating Compressible Media
	1 Introduction
	2 Bidimensional Models of Rotating Compressible Medium
	3 Non-localized Vortex
		3.1 Axisymmetric case
		3.2 Range of Instability
		3.3 Range of Stability
		3.4 Range of Possible Stability
	4 Localized Vortex
		4.1 Linear Analysis, Radial Perturbations
		4.2 ``Linear Profile Velocity'' Approximation
	5 Conclusion
	References
Coupled Scheme for Hamilton–Jacobi Equations
	1 Introduction
	2 Semi-lagrangian Schemes FF14
		2.1 Ultra-Bee Scheme for HJ Equations BZ07
	3 Construction of the Coupled Scheme (CS)
	4 Numerical Tests
	5 Conclusion and Future Work
	References
Compressible Heterogeneous Two-Phase Flows
	1 General Problems for the Modeling of Two-Phase Flows
		1.1 Averaged Models
		1.2 The Baer–Nunziato Model
	2 Analysis of Baer–Nunziato Type Models
		2.1 Basic Properties
		2.2 Entropy and Symmetric Form
	3 Mathematical Closure of Baer–Nunziato Type Models
		3.1 Interfacial Velocity
		3.2 Interfacial Pressure
		3.3 The Interfacial Wave and the Riemann Problem
	4 Dissipative Source Terms
		4.1 Derivation and Basic Properties
		4.2 Relaxation and Hierarchy of Models
		4.3 Nonlinear Stability
	5 Some Additional Remarks
	References
Bound-Preserving High-Order Schemes for Hyperbolic Equations: Survey  and Recent Developments
	1 Introduction
	2 Bound-Preserving First-Order Schemes
	3 Bound-Preserving High-Order Schemes
	4 Another Approach: Flux Correction
	5 Conclusions and Future Work
	References
Comparison of Shallow Water Models  for Rapid Channel Flows
	1 Shallow Water Models
	2 Numerical Experiments
		2.1 Initial and Boundary Conditions
		2.2 Numerical Solvers
		2.3 Numerical Results
	3 Discussion
		3.1 Water Level
		3.2 Shock Position
	4 Conclusion
	References
On Stability and Conservation Properties of (s)EPIRK Integrators in the Context of Discretized PDEs
	1 Introduction
	2 (s)EPIRK Schemes
		2.1 A-Stability
		2.2 CFL Condition
		2.3 Conservation
	References
Compactness on Multidimensional Steady Euler Equations
	1 Introduction
	2 A Heuristic Example
	3 Subsonic-Sonic Limits
	4 Incompressible Limits
	5 Application
	References
A Constraint-Preserving Finite Difference Method for the Damped Wave Map Equation to the Sphere
	1 Introduction
	2 The Numerical Method
		2.1 Discretization of the Domain and the Differential Operators
		2.2 Definition of the Finite Difference Scheme
	3 Convergence
	4 Numerical Examples
	References
Integral Transform Approach to Solving Klein–Gordon Equation with Variable Coefficients
	1 Introduction
	2 Linear Equations in the De Sitter Spacetime
	3 The Semilinear Equations in the De Sitter Spacetime
	References
Asymptotic Consistency of the RS-IMEX Scheme for the Low-Froude Shallow Water Equations: Analysis and Numerics
	1 RS-IMEX Schemes: An Introduction
	2 RS-IMEX Scheme for the Shallow Water Equations
	3 Main Result: Asymptotic Analysis of the Scheme
		3.1 Solvability
		3.2 Asymptotic Consistency
	4 Numerical Results
	References
Class of Space-Time Entropy Stable DG Schemes for Systems of Convection–Diffusion
	1 Introduction
	2 Space–Time Discontinuous Galerkin Scheme
		2.1 Convective Discretization
		2.2 Viscous Discretization
	3 Entropy Stability
	4 Numerical Results
		4.1 Scalar Advection-Diffusion
		4.2 Navier–Stokes Equations
	References
Invariant Manifolds for a Class of Degenerate Evolution Equations and Structure of Kinetic Shock Layers
	1 Introduction
		1.1 Equations and Assumptions
		1.2 Chapman–Enskog Expansion and Canonical Form
		1.3 Dichotomies Versus Direct Lp Estimate
		1.4 Results
		1.5 Discussion and Open Problems
	2 Reductions and Main Example
		2.1 Boltzmann's Equation
		2.2 Macro–Micro Decomposition
		2.3 Reduction to Canonical Form
	3 Linear Resolvent Estimates
		3.1 Symmetric Degenerate Evolution Equations
		3.2 Details/Counter Examples
		3.3 The Banach Space Setting
	4 H1 Stable Manifold Theorem
	5 Existence of a Center Manifold
	6 Structure of Small-Amplitude Kinetic Shocks
	References