Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy; A Liber Amicorum to Professor Godunov

Preface
Contents
Determination of Discontinuity Surfaces of Transport Equation Coefficients
	1 Introduction
	2 Direct Problem
	3 Inverse Problem
	References
Nonlocal Problems of Asymptotic Methods of Perturbation Theory
	1 Motivation of the Problem
	2 The Idea of Modifying the Averaging Method
	References
Numerical Solution of the Axisymmetric Dirichlet–Neumann Problem for Laplace\'s Equation (Algorithms Without Saturation)
	1 Introduction
	2 Statement of the Problem
	3 Discretization of Eq.(1) and the Statement of the Result
	References
Contributions to the Mathematical Technology Transfer with Finite Volume Methods
	1 Environmental Shallow Water Flows
	2 Low Mach Number Flows
	3 Gas Flow in Transport Networks
	References
An Easy Control of the Artificial Numerical Viscosity to Get Discrete Entropy Inequalities When Approximating Hyperbolic Systems of Conservation Laws
	1 Main Motivations
	2 Godunov-type Scheme with Artificial Viscosity
	3 Discrete Entropy Inequality
	4 Numerical Results
	References
Development of the Matrix Spectrum Dichotomy Method
	1 Contradictions Between the Classical Spectral Problem and Applied Problems
	2 New Statement of the Problem
	3 Base Algorithms
	4 Dichotomy Relative to the Second-Order Curves
	5 Applications
	6 Conclusion
	References
MHD Model of an Incompressible Polymeric Fluid. Stability of the Poiseuille Type Flow
	1 Introduction
	2 Nonlinear Model of the Polymeric Fluid Flow in a Plane Channel Under the Presence of an External Magnetic Field
	3 Periodic Perturbations. Formulation of Main Results
	References
Mathematical Models of Plasma Acceleration and Compression in Coaxial Channels
	1 Introduction
	2 Mathematical Model
	3 Plasma Acceleration in Presence of Longitudinal Magnetic Field
	4 Compression Plasma Flows in the Channels
	References
Pseudospectrum and Different Quality  of Stability Parameters
	1 Floating-Point Arithmetic
	2 Pseudospectrum and Spectral Portrait of a Matrix
	3 Hurwitz Stability Problem
	4 Schur Stability Problem
	References
Well-Balanced High-Order Methods  for Systems of Balance Laws
	1 Introduction
	2 Well-Balanced High-Order Finite Volume Scheme
	3 Well-Balanced Reconstruction Operators
	4 An Example
	5 Further Comments
	References
An All-Regime and Well-Balanced Lagrange-Projection Scheme for the Shallow Water Equations on Unstructured Meshes
	1 Governing Equations and Asymptotic Limit
	2 Acoustic-Transport Decomposition and Algorithm
	3 Numerical Experiments
	References
On the Dynamics of a Free Surface that Limits the Final Mass of a Self-Gravitating Gas
	1 Formulation of the Problem
	2 The Existence and Uniqueness Theorem for the Solution of the Cauchy Problem for an Integral Equation of Volterra Type
	3 An Example of the Evolution of the Free Surface of a Gaseous Sphere
	References
Exponential Stability of Solutions to Delay Difference Equations with Periodic Coefficients
	1 Delay Difference Equations
	2 Estimates for Solutions to Delay Difference Equations
	References
On Estimates of Solutions to One Class of Functional Difference Equations with Periodic Coefficients
	1 Introduction
	2 Criteria for Asymptotic Stability
	3 Estimates for Solutions to (1)
	References
Integral Operators at Settings and Investigations of Tensor Tomography Problems
	1 Main Equations and Uniqueness
	2 Angular Moments of Generalized ART
	3 Conclusion
	References
Lagrangian Godunov Schemes
	1 Seminal S.K. Godunov Ideas About Lagrangian Solvers
	2 Multidimensional Version
		2.1 Position of the Problem
		2.2 Discrete Objects
	3 Constraints
	References
On Numerical Methods for Hyperbolic PDE with Curl Involutions
	1 Introduction
	2 Model Problem and Different Approaches to Account for the Curl Involution
		2.1 SHTC Structure and Godunov–Powell Terms for Curl Involutions
		2.2 GLM Curl Cleaning
		2.3 An Exactly Curl-Free Discretization
		2.4 Numerical Results
	3 Conclusions
	References
Modeling Sodium Combustion with Liquid Water
	1 Introduction
	2 Multiphase Model and Properties
	3 Thermodynamic Closure
	4 Flow Solver
	5 Computational Example
	6 Conclusion
	References
Geodesic Mesh MHD: A New Paradigm for Computational Astrophysics and Space Physics Applied to Spherical Systems
	1 Introduction
	2 Mesh Construction and Representing Spherical Geometry
	3 Algorithm
	4 Results
		4.1 Accuracy Tests
		4.2 MHD Blast Problem
	5 Parallelism and Achieving Petascale Performance
	6 Discussion and Conclusion
	References
The Works of the S. K. Godunov School on Hyperbolic Equations
	References
On Two-Flow Instability of Dynamical Equilibrium States of Vlasov–Poisson Plasma
	1 Formulation of Problem in Kinetic Description
	2 Reformulation of Problem in Hydrodynamic Description
	3 A Priori Exponential Lower Estimate for Small 1D Perturbations Growing over Time
	4 Example
	References
Analytical Results for the Riemann Problem for a Weakly Hyperbolic Two-Phase Flow Model of a Dispersed Phase in a Carrier Fluid
	1 Introduction
	2 The Two-Phase Flow Model
		2.1 Equation of State
	3 The Dispersed Phase Equations
		3.1 The Case v- = v+=v
		3.2 The Case v- < v+
		3.3 The Case v- > v+
	4 The Carrier Phase Quantities
	5 Conclusion
	References
A Numerical Method for Two Phase Flows with Phase Transition Including Phase Creation
	1 Problem Formulation and Main Challenges
	2 Numerical Method
		2.1 Grid Alignment and Adaption
		2.2 Solution at the Interface
	3 Phase Creation
		3.1 Cavitation and Nucleation
	References
An Optimization Method for Feedback Stabilization of Linear Delay Systems
	1 Introduction
	2 Preliminaries
	3 Optimization Method for Feedback Stabilization
	References
Ten Good Reasons for Using Polyharmonic Spline Reconstruction in Particle Fluid Flow Simulations
	1 Introduction
	2 The Basic Reconstruction Problem of Particle Flow Simulation
	3 Kernel-Based Reconstruction Versus Polynomial Reconstruction
	4 Kernel-Based Reconstruction in Particle Fluid Flow Simulations
	5 Kernel-Based Reconstruction by Polyharmonic Splines
	6 Ten Good Reasons for Using Polyharmonic Spline Reconstruction
	References
Generalized Solutions of Galileo-Invariant Thermodynamically Consistent Conservation Laws, Constructed Using Ideas of Fundamental Works by S. K. Godunov
	1 Introduction
	2 Closed System of Quasi-linear Equations with Continuous Heat
	3 Thermodynamically Consistent Conservation Laws
	4 Conclusions
	References
Godunov-Type Schemes for Diffusion and Advection-Diffusion
	1 Introduction
	2 Diffusive Generalized Riemann Problem
	3 Advection Diffusion Generalized Riemann Problem
	4 Numerical Results
	5 Conclusion
	References
Direct and Inverse Problems  for Conservation Laws
	1 Introduction
	2 Direct Problem
	3 Inverse Problem of Recovering the Density
	References
On a Heat Wave for the Nonlinear Heat Equation: An Existence Theorem and Exact Solution
	1 Introduction
	2 Formulation
	3 Main Theorem
	4 Immobile Heat Wave
	References
Algorithm for Solving the Inverse Problem of 2-D Diffraction of a Linearly Polarized Elastic Transverse Wave for an N-Layer Medium with Composite Hierarchical Inclusions
	1 Introduction
	2 Main Result
	3 Conclusion
	References
Relaxation-Projection Schemes, the Ultimate Approximate Riemann Solvers
	1 Introduction
	2 Approximate Riemann Solvers
	3 Applications
	4 Conclusion
	References
New Efficient Preconditioner  for Helmholtz Equation
	1 Introduction
	2 Helmholtz Equation in a 3D Inhomogeneous Medium
	3 Solving the Helmholtz Equation by Iterative Methods
	4 Algorithm Implementation and Numerical Experiments
	References
Edge-Based Reconstruction Schemes  for Unstructured Meshes and Their Extensions
	1 Introduction
	2 Original and Simplified Vertex-Centred Formulations
	3 Cell-Centred Formulation
	4 EBR Schemes for Hybrid Unstructured Meshes
	5 Shock Capturing EBR Schemes
	6 Hybrid EBR Scheme for LES-Based Simulations
	7 Other Extensions of EBR Scheme
	8 Conclusion
	References
The Collision of Giant Molecular Cloud with Galaxy: Hydrodynamics, Star Formation, Chemistry
	1 Introduction
	2 Hydrodynamical Model of Galaxies
	3 Sub-grid Physics
		3.1 Star Formation and Supernova Feedback
		3.2 Cooling and Heating Functions
		3.3 Supermassive Black Holes
	4 The Chemistry
	References
Model Dirac and Dirac–Hestenes Equations as Covariantly Equipped Systems of Equations
	1 Complex Covariantly Equipped Systems of Equations of First Order
	2 Covariantly Equipped Systems of Equations in mathbbR1,n-1 as a Subset of the Set of Friedrichs Symmetric Hyperbolic Systems of Equations of First Order
	3 Modified Dirac–Hestenes Equation as a Covariantly Equipped System of Equations
	4 How to Reduce a Cauchy Problem for a Model Dirac Equations with Nonabelian Gauge Symmetry to a Cauchy Problem for Covariantly Equipped System of Equations
	References
Developments of the Godunov Method: Explicit–Implicit Scheme, Aeroacoustics
	1 Introduction
	2 Hybrid Implicit–Explicit Godunov Scheme
	3  Variational Riemann Problem and Aeroacoustics
	References
A Note on Construction of Continuum Mechanics and Thermodynamics
	1 Introduction
	2 Some Sources of Additional Insight into Modelling
	3 Conclusion
	References
A Method of Boundary Equations for Unsteady Hyperbolic Problems in 3D
	1 Introduction
	2 Description of the Method
	3 Numerical Demonstrations
	References
Solution of Deformable Solid Mechanics Dynamical Problems  with Use of Mathematical Modeling  by Grid-Characteristic Method
	1 Introduction
	2 Mathematical Model
	3 Results of Numerical Modeling
		3.1 Traumatic Brain Injuries
		3.2 Direct Problems of Exploration Seismology
		3.3 Impact Task
	References
Set of Solutions of Ordinary Differential Equations in Stability Problems
	1 Some Properties of Sets of Solutions of Ordinary Differential Equations
	2 Estimates of the Solutions of ODE Systems with a Change (Disturbance) in the Initial Data
	References
Two-Phase Computational Model  for Small-Amplitude Wave Propagation in a Saturated Porous Medium
	1 Introduction
	2 Thermodynamically Compatible Master System  for Two-Phase Saturated Porous Medium
	3 Small-Amplitude Waves in a Saturated Porous Medium
	4 Numerical Examples
		4.1 Dispersion Relation
		4.2 Two-Dimensional Wave Propagation
	References
Godunov Symmetric Systems and Rational Extended Thermodynamics
	1 Balance Laws, Entropy Principle, and Godunov Systems
	2 Rational Extended Thermodynamics and the Kinetic Theory
	3 Polyatomic Gas and Recent Results
	References
Variational Inequalities in the Dynamics of Elastic-Plastic Media: Thermodynamic Consistency, Mathematical Correctness, Numerical Implementation
	1 Introduction
	2 Principal Results
	3 Conclusions
	References
Estimates of Solutions to a System of Differential Equations with Two Delays
	1 Problem Statement
	2 Stability of Stationary Solutions
	3 Estimates of Solutions
	References
Moving-Coordinate Method and Its Applications
	1 Introduction
	2 Governing Equations of Flow Including Moving Boundaries
		2.1 Derivation of Conservative Form in General Coordinate System
		2.2 ALE (Arbitrary Lagrangean–Eulerian) Formulation
	3 Moving-Coordinate Method
	4 Summary
	References
Numerical Methods for Model Kinetic Equations and Their Application to External High-Speed Flows
	1 Introduction
	2 Governing Equations
	3 Basic Parallel Method of Solution
	4 Modifications to High-Speed Flows
	5 Example
	References
The ADER Path to High-Order Godunov Methods
	1 Introduction
	2 Godunov Methods and Riemann Problems
	3 Solvers for the Generalized Riemann Problem GRPk
		3.1 The Toro–Titarev GRPk solver
		3.2 The Montecinos–Toro Implicit GRP Solver
		3.3 Other GRPk Solvers
	4 Conclusions
	References
Near-Wall Domain Decomposition  for Modelling Turbulent Flows: Opportunities and Challenges
	1 Introduction
	2 Approximate Near-Wall Domain Decomposition
	3 Unsteady Near-Wall Domain Decomposition
	4 Exact Domain Decomposition
	5 Conclusions
	References
Quasilinear Integrodifferential Riccati-Type Equations
	1 Introduction
	2 Statement of the Problem and Main Results
	References
S.K. Godunov and Kinetic Theory in KIAM RAS
	References
Modeling the Explosive Phenomena Driven by Rapid Phase Transition
	1 Introduction
	2 Boiling Liquid Expanding Vapor Explosions
	3 Steam Explosions
	4 Conclusions
	References
Explicit-Iteration Scheme for Time Integration of the Navier–Stokes Equations
	1 Introduction
	2 The LI-M Scheme
	3 Time Integration LINS Scheme for the Navier–Stokes Equations
	References