On the Solution of Fredholm Integral Equations of the First Kind

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

Детонация горючей газовой смеси при взаимодействии ударной волны с эллиптической областью тяжелого инертного газа

On the basis of the Euler equations, the interaction of a shock wave in a combustible gas with an elliptical bubble of an inert gas of increased density is numerically simulated within a plane two-dimensional formulation. The finite-volume Godunov-type method of the second order of approximation is applied. Gas combustion is modeled using the Korobeinikov-Levin two-stage kinetics. Various values of the Mach number of the incident wave and the elongation of the inert bubble are considered, and the refraction and focusing of the incident shock are described. Qualitatively different regimes of gas detonation initiation have been found, including direct initiation by a strong wave, ignition upon reflection of an average-intensity wave from the gas interface, and upon focusing of secondary shock waves at lower shock Mach numbers. The dependence of the ignition mode on the shock intensity and the shape of the bubble is determined.

DETONATION INITIATION IN COMBUSTIBLE GAS MIXTURE UPON INTERACTION OF A SHOCK WAVE WITH AN INERT GAS BUBBLE

On the basis of the Euler equations for a perfect gas with an inhomo-geneous distribution of molar mass, the interaction of a shock wave in a combustible gas with an elliptic bubble of an inert gas of elevated density is simulated in planar two-dimensional (2D) formulation. Various values of the Mach number M of the incident wave and the ratio of bubble axles are considered. A finite-volume Godunov-type method of the second order of approximation with HLLC (Harten-Lax-van Leer Contact) Riemann solver is used for numerical simulation. The combustion reaction of the gas mixture is modeled using the two-stage Korobeynikov-Levin kinetics.

Comparison of a modified large-particle method with some high resolution schemes. Two-Dimensional test problems

Исследуются вычислительные свойства предложенной ранее новой модификации метода крупных частиц на основе нелинейной коррекции искусственной вязкости на первом (эйлеровом) этапе и гибридизации потоков на втором (лагранжевом и заключительном) этапе, дополненной двухшаговым алгоритмом РунгеКутты по времени. Метод обладает вторым порядком аппроксимации по пространству и времени на гладких решениях. На примере тестовых задач сверхзвукового потока газа в канале со ступенькой и двойного маховского отражения подтверждена работоспособность и вычислительная эффективность метода в сравнении с современными схемами высокой разрешающей способности. A number of computational properties of the previously proposed new modification of a largeparticle method are studied on the basis of a nonlinear correction of artificial viscosity at the first (Eulerian) stage and a hybridization of fluxes at the second (Lagrangian and final) stage supplemented by a twostep RungeKutta algorithm in time. The method has a second order of approximation in space and time on smooth solutions. The computational efficiency of the method is shown compared to several modern high resolution schemes using the forward facing step problem and the double Mach reflection problem.

Well-posedness of a nonlocal boundary value difference elliptic problem

The second order of approximation two-step difference scheme for the numerical solution of a nonlocal boundary value problem for the elliptic differential equation [see formula in PDF] in an arbitrary Banach space E with the positive operator A is presented. The well-posedness of the difference scheme in Banach spaces is established. In applications, the stability, almost coercive stability and coercive stability estimates in maximum norm in one variable for the solutions of difference schemes for numerical solution of two type elliptic problems are obtained.

MONOTONE AND CONSERVATIVE DIFFERENCE SCHEMES FOR ELLIPTIC EQUATIONS WITH MIXED DERIVATIVES

In the paper elliptic equations with alternating‐sign coefficients at mixed derivatives are considered. For such equations new difference schemes of the second order of approximation are developed. The proposed schemes are conservative and monotone. The constructed algorithms satisfy the grid maximum principle not only for coefficients of constant signs but also for alternating‐sign coefficients at mixed derivatives. The a prioriestimates of stability and convergence in the grid norm C are obtained.

Monotone Schemes of a Higher Order of Accuracy for Differential Problems with Boundary Conditions of the Second and Third Kind

In the present paper monotone difference schemes of the second order of approximation and accuracy for differential boundary-value problems of the second and third kind without using the basic differential equation at the domain of the boundary are constructed. The main idea is based on the assumption of the existence and uniqueness of a smooth solution in some sufficiently small neighborhood of the definition domain of the problem and the use of only half-integer nodes of the grid (boundary points are excluded from the calculated nodes). In this case, the boundary conditions are directly approximated with the second order on a two-point stencil. If we assume that the equation has a meaning at the boundary nodes as well, then in this case monotone schemes of the fourth order of accuracy have been constructed. It is shown that in the case of Neumann problem it is necessary to construct such computational procedures, which are monotone and satisfy the grid maximum principle with respect to the flow (of the first derivatives with respect to space variables).