Vol 60, No 7 (2024)

The paper studies the symmetric eigenvalue problem with nonlinear dependence on the spectral parameter in a Hilbert space which is a vector lattice with a cone of positive elements. The existence of a positive simple minimum eigenvalue corresponding to a single normalised positive eigenelement is established. The approximation of the problem in a finite-dimensional subspace is investigated. Results on the convergence and error of approximations to the minimum eigenvalue and the corresponding positive eigenelement are obtained. Computational methods for solving matrix eigenvalue problems with nonlinear dependence on the spectral parameter are developed and justified. The results of numerical experiments illustrating theoretical conclusions are given.

The work is devoted to the derivation and numerical studies of a difference scheme with well-controlled dissipation for solution of equations of the Kapila model. Kapila model is widely used for analysis of two-phase compressible flows. It has a form of first order non-conservative hyperbolic system. As any other 1st order non-conservative hyperbolic system it requires definition of the regularizing dissipative operator to define discontinuous solutions and Rankin–Hugoniot conditions. The choice of dissipative operator influence wave structure observed in the solutions. Schemes with well-controlled are constructed in such a way that the dissipative operator which is determined by the form of their equivalent equation coincides with the one used to define correct setting of the original problem to be solved. As a result, it is expected that numerical solution converges to the solution of the system under consideration. Numerical experiments presented in the work demonstrate the effectiveness of this approach. As exact solutions numerical solutions of the traveling wave type obtained by other methods were used.

The conservative finite volume scheme for heat transfer problem in two-dimensional region with moving boundaries is presented. The two-phase Stefan problem is considered as an example. To track the moving interface between solid and liquid, the front-fixing technique is applied. The time varying physical domain is mapped to a fixed computational space with regular boundaries. Finite volume approximation of governing equations is constructed in computational domain on fixed rectangular grid. The geometric conservation law is incorporated into the numerical scheme. The Jacobian and the grid velocities of the control volume are evaluated to satisfy the discrete form of the Jacobian transport equation. This procedure guarantees the enforcing of space conservation law in the physical domain. The numerical scheme inherits the basic properties of the original differential problem.

A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized version of spline method are proposed and justified. The optimality in the order of accuracy of the method constructed is established.