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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

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.

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.

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.