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The ideas of formulating a weak solution for a hyperbolic system of one-dimensional gas dynamics equations are presented. An important aspect is the examination of the scheme for the fulfillment of the nondecreasing entropy law, which must hold for weak solutions and is obligatory from a physics point of view. The concept of a weak solution is defined in a finite-difference formulation with the help of the simplest linearized version of the classical Godunov scheme. It is experimentally shown that this version guarantees an entropy nondecrease. As a result, the growth of entropy on shock waves can be simulated without using any correction terms or additional conditions.

A new formulation of the Godunov scheme with linear Riemann problems is proposed that guarantees a nondecrease in entropy. The new version of the method is described for the simplest example of one-dimensional gas dynamics in Lagrangian coordinates.

This paper describes the author’s thoughts and reflection inspired after publishing paper [1] and com� paring it with the lecture [2] delivered at a conference in Lyon. This lecture was based on coauthored work [3]. In [1] the equations of relativistic fluid dynamics of a charged liquid dielectric were supplemented with Maxwell’s equations for the electromagnetic field produced by this charge. Primary attention was given to the overcoming of the difficulties associated with the relativistically invariant incorporation of the param� eters μ and e depending on hydrodynamic variables, specifically, on the velocity. The study was a develop� ment of Minkowski’s classical work [4]. More specifically, a Lorentz invariant W(E, B, μ, e) that is a qua� dratic form in B i and E k was used to write Maxwell’s equations as

The results of numerical and experimental simulation of wave formation under an oblique impact of metal plates during explosion welding are presented. The numerical simulation was carried out on the basis of the Maxwell relaxation model and by a molecular dynamics method. In the experiments, the impact of metal plates was investigated with X-ray radiography of phenomena in front of and behind the point of contact, and the preserved samples were studied metallographically. It is shown that the numerical simulation correctly reproduces the formation and evolution of waves on the contact boundary. Simultaneously, constraints are pointed out that prohibit the use of the elastoplastic model in the impact zone of the plates starting from the moment when the material of the plate in this zone is decomposed into thin jets. In this zone, the dependence of the specific deformation energy E on the tensor Cji is no longer described by a convex function. It is this fact that motivated the transition to the molecular dynamics model.

Minkowski’s classical work underlying modern electrodynamics is described. Primary attention is given to the mathematical refinements that are required if the parameters ɛ and μ depend on the properties of the dielectric fluid, i.e., the medium carrying charges in the field under study. It is shown that the motion of the medium and the accompanying evolution of the electromagnetic field are described by differential equations that are symmetric and hyperbolic in the sense of Friedrichs. This property guarantees their well-posedness. Note that this class of equations was not known in Minkowski’s time. At present, it plays an important role in the mathematical simulation of nonstationary processes and in the design of numerical algorithms. The author’s view of the mathematical foundations of Minkowski’s work is presented, which relates the latter to present-day insights into the theory of differential equations. This paper can possibly be of interest to physicists.

Convergence of the approximate solution of fluid dynamics problems obtained using Godunov’s scheme to the discontinuous solution is investigated.

The paper is devoted to new applications of the ideas underlying Godunov’s method that was developed as early as in the 1950s for solving fluid dynamics problems. This paper deals with elastoplastic problems. Based on an elastic model and its modification obtained by introducing the Maxwell viscosity, a method for modeling plastic deformations is proposed.

The work is devoted to some important questions that come with the solution of gas dynamics equations using standard Godunov scheme with the corrections of A V Safronov. The numerical solution is succeeded by intrinsic differential realization of energy conservation law. It has been found experimentally that in all computational cells the entropy nondecreasing is provided. The fact makes it possible to model the entropy rising on shock waves. Besides the experiments described in the paper gives the intrinsic explanation of the reasons for the appearance of the zones with decreased accuracy order in the results. The influence of the computational grid parameters (Courant number) on the plots of grid structures of shock waves is also studied.

We describe a method for representing the nonlinear system of gas dynamics equations in quasilinear form with symmetric coefficient matrices and, moreover, with a positive definite matrix at the time derivative.

Let \( \tilde \lambda \) be an approximate eigenvalue of multiplicity mc = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting mc rows of T−\( \tilde \lambda \)I so that the condition number of the r × n matrix B formed of the remaining r rows is as small as possible. A special basis of mc vectors with local supports is constructed for the subspace kerB. These vectors are approximate eigenvectors of T corresponding to \( \tilde \lambda \). Another method for deleting mc rows of T−\( \tilde \lambda \)I is also proposed. This method uses a rank-revealing QR decomposition; however, it requires a considerably larger number of arithmetic operations. For the latter algorithm, the condition number of B is estimated, and orthogonality estimates for vectors of the special basis of kerB are derived.

The basic results are overviewed concerning the formulation of nonlinear elasticity equations in the form of symmetric hyperbolic systems in long-time studies performed under the direction of the first author. The underlying principles developed in those studies are stated, and some inaccuracies and errors are corrected. Procedures are described for computing the coefficient matrices of the equations from a given equation of state.

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.

An algorithm is constructed for calculating invariant subspaces of symmetric hyperbolic systems arising in electromagnetic, acoustic, and elasticity problems. Discrete approximations are calculated for subspaces that correspond to minimal eigenvalues and smooth eigenfunctions. Difficulties related to the presence of an infinite-dimensional kernel in the differential operator are successfully handled. The efficiency of the algorithm is demonstrated using acoustics equations.

The numerical analysis of the spectral structure of symplectic matrices is proposed. The strong stability of symplectic matrices and linear Hamiltonian systems with periodic coefficients is discussed. Applications to the theory of parametric resonance are illustrated by spectral portraits calculation.

Formalization of the evolutionary equations of continuum mechanics in the form of a Galilean-invariant nondivergent hyperbolic system is described. Special attention is paid to supplementing the system by additional equations required for validity of the conservation laws. A new version of Maxwell relaxation terms is proposed which is consistent with the additional equations and ensures gauge invariance.

The hydrodynamics of a multiphase media or a mixture is formalized in the form of a hyperbolic system taking into account chemical reactions. Additional conditions consistent with the system are found for the solutions to ensure conservation of energy and momentum.

A thermodynamically consistent Galilean-invariant formalization of the elasticity equations is proposed. The equations under consideration form a symmetric hyperbolic system but they are non-conservative. The conservation laws are valid only on a solution with special initial values. Difficulties arising while introducing the dissipative terms are briefly discussed.

We calculate the Clebsch–Gordan coefficients for orthogonal rather than unitary representations of the rotation group.

This paper continues previous investigation of Galilean‐invariant equations of mathematical physics, which was begun with the use of the Clebsch–Gordan coefficients from the theory of products of the represented group SO(3). Complex systems of conservation laws and thermodynamic identities are constructed. Concrete examples are given.

This paper gives an introduction to formalization of Galilean‐invariant and thermodynamically consistent equations of mathematical physics in which unknowns are transformed in rotations by irreducible representations of integer weights. This formalization is based on the theory of representations of the group SO(3).

We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.

We propose a generalized Krylov subspace method for computing the invariant subspace associated with the smallest eigenvalues of an elliptic operator. The success of the method relies upon a smoothing operator whose main purposes is to enhance the components of the smallest eigenvalues while dumping those of the largest ones. We show numerically that multigrid techniques can be used to construct an efficient smoothing operator.

A variational principle for grid generation is described. A unique solution of the variational problem is represented by a quasi-isometric mapping of the unit square onto a physical domain. The variational principle is based on the theory of conformal mapping and the geometry of the surface of constant curvature.

A discussion of the author's advanced viewpoint on the underlying prin- ciples for construction of Godunov's scheme is presented. The application of these principles in problems of elastic and elastoplastic deformations is outlined. The pre- sentation is based on extensive numerical simulations performed for both an analysis of the solution convergence details with decreasing mesh step size (for equations of Fluid Dynamics) and the motivation of modeling an elastoplastic media by an "effective elastic deformation" and the employment of Maxwell's viscosities. Being engaged for 58 years in gas dynamics problems and in more general ones of continuum mechanics, also developing algorithms for numerical simulation of unsteady phenomena, I understood the need to directly construct discrete models for a media composed of elementary microcells rather than to rely only on commonly used differential equations available in monographs and textbooks. In this way, one should aim at obtaining such a behavior of the macro objects (composed of the microcells) that would realistically enough resemble the phenomena in the actual media to be approximated by our model. In fact, exactly this kind of modeling had been performed in "Godunov's scheme" in 1953-1954 (published in 1959), though the above interpretation of the scheme was elicited some later. In the current lecture, I would like to deliver my up-to-date understanding of the scheme and describe its extension to problems of elastoplastic deformations, which has been developed together with my colleagues. This lecture can be viewed as a sequel of the one given at Michigan University in May 1997 and published in Novosibirsk (5). A condensed version of the paper (5) came out in English in J. Comp. Phys. (6), whereas a translation of the whole paper into English was published by INRIA in 2008 (7). The overdetermination of equations governing smooth solutions turned out to be of crucial importance. At the same time, for modeling discontinuities of solutions

A method for calculating principal invariant subspaces of symmetric hyperbolic systems arising in fluid dynamic, electromagnetic and elasticity problems is presented. Such subspaces correspond to a certain set of minimal nonzero eigenvalues and contain sufficiently smooth functions, low-frequency modes. Difficulties related to the presence of an infinite-dimensional null space in the differential operator are successfully handled. The efficiency of the algorithm is demonstrated for acoustics equations.

The quasi-conformal, quasi-isometric technique is described for the block-structured grid generation. The grid is constructed by minimization of the variational functional. The feature of computational implementation is presented.

The calculation of eigenvalues of matrices is the classical problem of linear algebra. Construction of standard programs for its solution comes across serious difficulties if matrices are non-symmetric. One can easily construct examples of matrices of not very high order whose spectrum, being calculated by computer, fills two-dimensional domains of a complex plane. In the automatic regulation theory and in the stability theore one often has to find out whether the whole spectrum lies in the left-hand half-plane or not (Hurwitz problem). Described and grounded in the paper is a computably stable algorithm for the solution of this problem, requiring no calculations of either eigenvalues or Hurwitz determinants.