Volume 19, Issue 6

We consider two known methods for increasing the order of a deterministic error in the Euler scheme and a new method based on the simultaneous estimation of a desired functional with two different time steps and the corresponding extrapolation. We propose to use the new method in the weight modification of the Euler scheme for the calculation of the gradient of the solution to a stationary diffusion equation. The results of numerical experiments are given.

We consider two known methods for increasing the order of a deterministic error in the Euler scheme and a new method based on the simultaneous estimation of a desired functional with two different time steps and the corresponding extrapolation. We propose to use the new method in the weight modification of the Euler scheme for the calculation of the gradient of the solution to a stationary diffusion equation. The results of numerical experiments are given.

To stabilize solutions of the Stokes and Navier–Stokes equations with a given rate by the boundary conditions we construct a discrete analogue of the algorithm proposed for the differential case by A. V. Fursikov. We carry out a complete computational stabilization cycle and discuss the differences from the continuous process.

To stabilize solutions of the Stokes and Navier–Stokes equations with a given rate by the boundary conditions we construct a discrete analogue of the algorithm proposed for the differential case by A. V. Fursikov. We carry out a complete computational stabilization cycle and discuss the differences from the continuous process.

Mathematical models for investigating the formation and propagation of hybrid combustion waves in the stationary layer of a catalyst are developed. In spherical and cylindrical symmetry reactors, the linear velocity of a flow is a radial variable. Therefore, the heat and mass transfer coefficients and the heat conductivity coefficient of the charge, which vary with the reactor radius, are taken into account in the mathematical model. The nonlinear dependence of the heat conductivity coefficient on reemission is also taken into account. To numerically calculate these regimes a special algorithm with an adaptive space-time grid in the gas-phase combustion zone is developed. The proposed algorithm allows us to effectively carry out the numerical investigation of the dynamical characteristics of hybrid waves in the combustion zone. We simulate the process of reaching the stationary mode in a spherical reactor and obtain the dependence of the position of the standing wave on the gas flow rate.

Mathematical models for investigating the formation and propagation of hybrid combustion waves in the stationary layer of a catalyst are developed. In spherical and cylindrical symmetry reactors, the linear velocity of a flow is a radial variable. Therefore, the heat and mass transfer coefficients and the heat conductivity coefficient of the charge, which vary with the reactor radius, are taken into account in the mathematical model. The nonlinear dependence of the heat conductivity coefficient on reemission is also taken into account. To numerically calculate these regimes a special algorithm with an adaptive space-time grid in the gas-phase combustion zone is developed. The proposed algorithm allows us to effectively carry out the numerical investigation of the dynamical characteristics of hybrid waves in the combustion zone. We simulate the process of reaching the stationary mode in a spherical reactor and obtain the dependence of the position of the standing wave on the gas flow rate.

This paper deals with methods for the solution of boundary value problems for linear systems of stochastic differential equations. We investigate numerical algorithms, the problem of existence and uniqueness of solutions and other specific problems (including stationary boundary value problems, reduction of a boundary value problem to a Cauchy problem, extended boundary value problems, active and passive boundary conditions, etc.).

This paper deals with methods for the solution of boundary value problems for linear systems of stochastic differential equations. We investigate numerical algorithms, the problem of existence and uniqueness of solutions and other specific problems (including stationary boundary value problems, reduction of a boundary value problem to a Cauchy problem, extended boundary value problems, active and passive boundary conditions, etc.).

In this paper we first formulate and prove a number of theorems concerning the convergence of combined numerical one-step methods for index 1 differential-algebraic systems. Then, we use these results to justify an implicit extrapolation technique and show their practical importance. Second, we give a theory of adjoint and symmetric one-step methods for differential-algebraic equations and we also determine symmetric methods among Runge–Kutta formulae. We prove that algebraically stable symmetric Runge–Kutta formulae are symplectic and they have a structure which is in some sense similar to the structure of Gauss methods. Finally, we come to the concept of quadratic extrapolation for index 1 differential-algebraic systems and develop an advanced version of the localglobal step size control based on the extrapolation technique. Numerical tests support the theoretical results of the paper.

In this paper we first formulate and prove a number of theorems concerning the convergence of combined numerical one-step methods for index 1 differential-algebraic systems. Then, we use these results to justify an implicit extrapolation technique and show their practical importance. Second, we give a theory of adjoint and symmetric one-step methods for differential-algebraic equations and we also determine symmetric methods among Runge–Kutta formulae. We prove that algebraically stable symmetric Runge–Kutta formulae are symplectic and they have a structure which is in some sense similar to the structure of Gauss methods. Finally, we come to the concept of quadratic extrapolation for index 1 differential-algebraic systems and develop an advanced version of the localglobal step size control based on the extrapolation technique. Numerical tests support the theoretical results of the paper.