Том 229, № 3 (2018)

We develop a general algorithm for the analysis the spectral stability of generalized multistage Runge–Kutta methods of various orders of accuracy as applied to the numerical integration with respect to time of an initial-boundary-value problem for the second-order parabolic equation. The expression for the function of spectral stability is obtained in two alternative forms: on the basis of matrix relations and in the determinant form. We study specific realizations of various generalized explicit Runge–Kutta methods and their spectral stability. It is shown that all explicit generalized Runge–Kutta methods possess the property of conditional spectral stability and the property of conditional monotonicity of the numerical solution in time whose violation leads to the appearance of false oscillations of the approximate solution. The stability function for these methods is polynomial. It is shown that the application of two-stage generalized explicit Runge–Kutta methods leads to the appearance of predictor-corrector-type schemes. In the case of nonstationary one-dimensional heat-conduction problem, on the basis of a one-stage generalized Runge–Kutta method, we obtain the classical two-layer conditionally stable explicit finite-difference scheme on a four-point template. It is demonstrated that the five-stage generalized Runge–Kutta–Merson method is characterized by the weakest condition of spectral stability as compared with all investigated explicit generalized Runge–Kutta methods.

We consider the problem of investigation of the spectrum of natural vibrations of a nonthin orthotropic shallow shell variable in two coordinate directions of thickness in the nonclassical statement. The approach to the solution of the obtained two-dimensional boundary-value problem is based on its reduction (by the method of spline-approximation of the unknown functions along one coordinate direction) to the one-dimensional problem with its subsequent solution. We study different cases of boundary conditions imposed on the contours of the shell. We also perform the comparison and analysis of the natural frequencies and modes of vibrations of orthotropic shells of constant and variable thickness.

We formulate and solve the problem of biaxial bending of an isotropic plate with two coaxial through cracks of identical lengths by distributed bending moments applied at infinity under the action of external load symmetric about the cracks with regard for the contact zone of their faces and in the presence of plastic zones near the crack tips, where the Tresca plasticity conditions are satisfied in the form of a surface layer or a plastic hinge. By using complex potentials of the plane problem and the classical theory of bending of the plates, we obtain an analytic solution of the problem in the class of functions bounded in the vicinity of the vertices of the plastic zones. The length of the plastic zones and the crack-tip opening displacements are found numerically.

The concept of local mass displacements (as a measure of nonlocality of the state of a physically small element of a body) is used to formulate a complete system of relations of the gradient-type mathematical model of viscous compressible fluid. In this model, the relationship between the thermomechanical processes and the process of local mass displacements, its irreversibility, and the inertia properties are taken into account. It is shown that, in the stationary approximation, the obtained system of relations enables us to justify the appearance of wedging pressure in thin interlayers of fluid.