Table Of ContentGennadii V. Demidenko
Evgeniy Romenski
Eleuterio Toro
Michael Dumbser Editors
Continuum
Mechanics, Applied
Mathematics and
Scientific Computing:
Godunov’s Legacy
A Liber Amicorum to Professor Godunov
Continuum Mechanics, Applied Mathematics
fi ’
and Scienti c Computing: Godunov s Legacy
Gennadii V. Demidenko Evgeniy Romenski
(cid:129) (cid:129)
Eleuterio Toro Michael Dumbser
(cid:129)
Editors
Continuum Mechanics,
Applied Mathematics
fi
and Scienti c Computing:
’
Godunov s Legacy
A Liber Amicorum to Professor Godunov
123
Editors
Gennadii V.Demidenko Evgeniy Romenski
SobolevInstitute ofMathematics SobolevInstitute ofMathematics
Novosibirsk, Russia Novosibirsk, Russia
Eleuterio Toro Michael Dumbser
Laboratory of Applied Department ofCivil, Environmental
Mathematics, DICAM andMechanicalEngineering
University of Trento University of Trento
Trento, Italy Trento, Italy
ISBN978-3-030-38869-0 ISBN978-3-030-38870-6 (eBook)
https://doi.org/10.1007/978-3-030-38870-6
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Preface
July 17, 2019 marked the 90th anniversary of the outstanding scientist, one
of the leaders of modern applied mathematics, Sergei Konstantinovich Godunov.
S.K.GodunovbelongstothatgenerationofSovietpeoplewhoatthecostofhuge
effortsrestoredtheSovietUnionafterthehardestlossessufferedduringtheSecond
World War. In the postwar years, the leadership of the Soviet Union paid special
attention to nuclear and space projects, the success of which gave a chance to
preserve the peace and independence of the country. S. K. Godunov was one of
those scientists who actively participated in these projects.
In1951aftergraduatingfromMoscowStateUniversity(MSU),S.K.Godunov
was accepted into the Calculation Bureau of the Steklov Mathematical Institute
of the USSR Academy of Sciences. Outstanding scientists B. N. Delaunay and
I.G.PetrovskiiwerehisscientificadvisersatMSU.Fromtheveryfirstdays,Sergei
Godunov took an active part in the work of the team of scientists created to solve
the most important practical problems in mathematical modeling and carry out
calculations of various processes in the field of nuclear physics. The mathematical
problems that needed to be solved arose as a result of discussions of renowned
mathematicians and physicists. Yu. B. Khariton, Ya. B. Zeldovich, I. E. Tamm,
A. D. Sakharov, E. I. Zababakhin, D. A. Frank-Kamenetskii, and other scientists
came to the Institute from the scientific centers of Sarov and Snezhinsk to discuss
the results of experiments and problem statements.
In1953,SergeiGodunovwasgivenataskbyM.V.KeldyshandI.M.Gelfand
tocreateavariantofthenumericalmethodofJ.NeumannandR.Richtmyerforthe
gas dynamics equations using artificial viscosity. However, the young scientist
proposed his own method and invented a difference scheme, which over time
gained worldwide fame as the “Godunov scheme”. In 1954, he defended his can-
didate dissertation, the basis of which was precisely this method. This dissertation
of S. K. Godunov was discussed by world-famous scientists I. M. Vinogradov,
I. M. Gelfand, M. V. Keldysh, I. G. Petrovskii, S. L. Sobolev, and many more.
In the following years, “Godunov scheme” had a huge impact on the develop-
ment of modern computational mathematics. Nowadays, there is a variety of
numerical methods for different classes of nonlinear equations of mathematical
v
vi Preface
physics and continuum mechanics. Many of them, including the “Godunov
schemes”ofhighorderofaccuracy,arebasedontheinitialideasofS.K.Godunov.
A significant role in the development of modern computational mathematics was
also played by the monograph “Introduction to the theory of difference schemes”,
written by S. K. Godunov and V. S. Ryabenkii (1962).
During his work at the Mathematical Institute and the Institute of Applied
Mathematics of the USSR Academy of Sciences (since 1966), S. K. Godunov
created many methods for approximate solving various problems of continuum
mechanics and laid the foundations of new areas of applied mathematics. Here are
justafewexamples.Tosolvenumericallytwo-dimensionalproblemsofcontinuum
mechanics in domains with complex boundaries, special requirements for compu-
tational grids arise. The problem of automating the construction of curvilinear
differencegridswasformulatedbyS.K.Godunov,andvariousmethodsforsolving
it weredeveloped under his leadership.S. K.Godunovcreatedthemethod oftime
marching to steady state for the numerical analysis of the gas flow around bodies.
S. K. Godunov created and justified the method of orthogonal successive substi-
tution for solving boundary value problems for systems of ordinary differential
equations.
Significant results were obtained by S. K. Godunov in the theory of quasilinear
equations. He studied the problem of uniqueness of generalized solution to gas
dynamicsequationsandconsideredquestionsontheplaceofcontinuummechanics
equationsinthetheoryofhyperbolicequationsinconservativeform,aswellasthe
generalizationoftheconceptofentropy,thelawofitsincreaseandthermodynamic
relations. The obtained results formed the basis of his doctoral dissertation (1965).
His work “An interesting class of quasilinear systems” (1961) on the connection
between the laws of thermodynamics and the well-posedness of problems for
models of continuum mechanics gave rise to a new direction of research of
hyperbolic systems of conservation laws in mathematical physics.
In 1969, S. K. Godunov moved to Novosibirsk at the invitation of
M. A. Lavrentyev and was appointed the head of the laboratory at the Computer
CenteroftheSiberianBranchoftheUSSRAcademyofSciences.Hisfirstworkin
Siberia was related to the study of the behavior of metals in explosion welding,
which was carried out jointly with experimental physicists from the Institute of
Hydrodynamics of the Siberian Branch of the USSR Academy of Sciences. To
describe the high-speed deformation of metals, S. K. Godunov proposed a new
approach to the modeling continuous media, which led to the creation of a unified
model of continuous medium capable of describing its elastic, plastic, and liquid
stateusingasinglesystemofgoverningdifferentialequations.Numericalmodeling
carried out under his leadership allowed to predict a new effect such as the for-
mation of a submerged jet when metal plates collide.
In 1980 at the invitation of S. L. Sobolev, he began to work in the Institute of
Mathematics of the Siberian Branch of the USSR Academy of Sciences. At the
Institute S. K. Godunov together with his disciples carried out active research of
mixedproblemsforhyperbolicsystems,computationalproblemsinstabilitytheory,
anddichotomyforordinarydifferentialequationsandinlinearalgebra.Despitethe
Preface vii
apparent differences, these areas are interrelated and are the continuation of the
workcarriedoutbyS.K.GodunovduringtheMoscowperiodofhisactivity.Many
neworiginalandunexpectedresultshavebeenobtainedinalltheseareas.Thishas
always been a characteristic feature in the scientific work of the Godunov school.
Here are just a few examples.
In the theory of partial differential equations, the problem of symmetrization of
two-dimensional strictly hyperbolic equations was solved, complete description
oftheenergyintegralswasgiven,andnewcriteriaforthewellposednessofmixed
problems for some classes of hyperbolic equations were obtained in terms of
coefficients.
Incomputationallinearalgebra,S.K.Godunovforthefirsttimeformulatedthe
concept of guaranteed accuracy, thereby laying the foundations of a new direction
in computational mathematics. He together with his disciples created a number of
new computational algorithms with guaranteed accuracy for solving the spectral
problem for non-Hermitian matrices, as well as for solving systems of linear
equations on a computer. Within the framework of the concept of guaranteed
accuracy, new fundamental concepts were introduced to ensure the rigor of com-
puter calculations: e-spectrum, spectral portrait, dichotomy quality criterion, etc.
Inthetheoryofordinarydifferentialequations,S.K.Godunovtogetherwithhis
disciples developed new algorithms that allow carrying out numerical studies of
asymptotic stability of stationary solutions to autonomous differential equations
with guaranteed accuracy.
Based on the developed mathematical apparatus, S. K. Godunov together with
colleagues and disciples actively continues developing algorithms and carries out
numericalanalysisofvariousmodelsofcontinuummechanics,sometimesreturning
to the problems considered earlier and carrying out the research at higher level.
Sergei Konstantinovich Godunov is selflessly devoted to science. He is always
full of energy, ideas, and creative plans. His conviction, high culture, and broad
erudition always attract colleagues, disciples, and followers to him.
OnAugust4–10,2019,inAkademgorodokofNovosibirsk(Russia),itwasheld
amajorinternationalconferenceinhonorofthe90thbirthdayofS.K.Godunov.It
was attended by more than 300 scientists from 24 countries and 34 regions of
Russia. Such great interest in the conference is undoubtedly connected with the
scale of the personality of Sergei Konstantinovich.
At the Opening Ceremony of the conference, S. K. Godunov gave the plenary
lecture “Memories of difference models in hydrodynamics”. On this day, invited
lectures were also given by eight leading Russian and foreign scientists. Each
of them somehow mentioned the ideas, results, and methods of S. K. Godunov.
Thiscanbeconcludedbythetitlesofthelectures:“OnS.K.Godunov’sworkson
theuraniumproblem”(Yu.N.Deryugin,R.M.Shagaliev,RussianFederalNuclear
Center, Sarov, Russia); “Godunov symmetric systems and rational extended ther-
modynamics” (T. Ruggeri, University of Bologna, Bologna, Italy); “Godunov
methods” (E. F. Toro, University of Trento, Trento, Italy), etc.
viii Preface
Therangeoftopicsthatwerewithinthescopeoftheconferenceincludedpartial
differential equations, equations of mathematical physics, differential-difference
equations, mathematical modeling, difference schemes, computational methods of
linear algebra, mathematical questions of gas dynamics, hydrodynamics, aerody-
namics, and some other sections of continuum mechanics. Leading specialists in
applied mathematics and mechanics from different countries of the world gave
invitedlecturesattheconference.Therewere42invitedlecturesandmorethan200
short communications and poster presentations.
It is gratifying that young researchers took an active part in the conference. All
of them had a unique opportunity to receive lessons from one of the classics of
modern applied mathematics S. K. Godunov, his closest colleagues, disciples, and
other famous scientists.
Detailed information about the conference is available at: http://www.math.nsc.
ru/conference/gsk/90/en.
The present book contains selected articles written on the materials of the talks
presented at the conference.
Novosibirsk, Russia Gennadii V. Demidenko
Novosibirsk, Russia Evgeniy Romenski
Trento, Italy Eleuterio Toro
Trento, Italy Michael Dumbser
Contents
Determination of Discontinuity Surfaces of Transport Equation
Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
E. Yu. Balakina
Nonlocal Problems of Asymptotic Methods of Perturbation Theory. . . . 7
V. S. Belonosov
NumericalSolutionoftheAxisymmetricDirichlet–NeumannProblem
for Laplace’s Equation (Algorithms Without Saturation). . . . . . . . . . . . 13
V. N. Belykh
Contributions to the Mathematical Technology Transfer with Finite
Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A. Bermúdez, S. Busto, L. Cea and M. E. Vázquez-Cendón
An Easy Control of the Artificial Numerical Viscosity to Get Discrete
Entropy Inequalities When Approximating Hyperbolic Systems
of Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Christophe Berthon, Arnaud Duran and Khaled Saleh
Development of the Matrix Spectrum Dichotomy Method . . . . . . . . . . . 37
Elina A. Biberdorf
MHD Model of an Incompressible Polymeric Fluid. Stability
of the Poiseuille Type Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A. M. Blokhin and D. L. Tkachev
Mathematical Models of Plasma Acceleration and Compression
in Coaxial Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
K. V. Brushlinskii and E. V. Stepin
Pseudospectrum and Different Quality of Stability Parameters . . . . . . . 61
H. Bulgak
ix
x Contents
Well-Balanced High-Order Methods for Systems of Balance Laws . . . . 69
Manuel J. Castro and Carlos Parés
An All-Regime and Well-Balanced Lagrange-Projection Scheme
for the Shallow Water Equations on Unstructured Meshes . . . . . . . . . . 77
Christophe Chalons, Samuel Kokh and Maxime Stauffert
On the Dynamics of a Free Surface that Limits the Final Mass
of a Self-Gravitating Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
N. P. Chuev
Exponential Stability of Solutions to Delay Difference Equations
with Periodic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
G. V. Demidenko and D. Sh. Baldanov
On Estimates of Solutions to One Class of Functional Difference
Equations with Periodic Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
G. V. Demidenko and I. I. Matveeva
Integral Operators at Settings and Investigations of Tensor
Tomography Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Evgeny Derevtsov, Yuriy Volkov and Thomas Schuster
Lagrangian Godunov Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bruno Després
On Numerical Methods for Hyperbolic PDE with Curl Involutions. . . . 125
M. Dumbser, S. Chiocchetti and I. Peshkov
Modeling Sodium Combustion with Liquid Water. . . . . . . . . . . . . . . . . 135
Damien Furfaro, Richard Saurel, Lucas David and François Beauchamp
Geodesic Mesh MHD: A New Paradigm for Computational
Astrophysics and Space Physics Applied to Spherical Systems. . . . . . . . 145
Sudip K. Garain, Dinshaw S. Balsara and Vladimir Florinski
The Works of the S. K. Godunov School on Hyperbolic Equations . . . . 153
Valeriĭ M. Gordienko
On Two-Flow Instability of Dynamical Equilibrium States
of Vlasov–Poisson Plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Yuriy G. Gubarev
AnalyticalResultsfortheRiemannProblemforaWeaklyHyperbolic
Two-Phase Flow Model of a Dispersed Phase in a Carrier Fluid. . . . . . 169
Maren Hantke, Christoph Matern and Gerald Warnecke
A Numerical Method for Two Phase Flows with Phase Transition
Including Phase Creation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Maren Hantke and Ferdinand Thein
