Estimates for the accuracy of the projection-difference method for an abstract hyperbolic-parabolic system like systems of thermoelasticity equations

We obtain energy estimates for the accuracy of a three-layer scheme of the projection- difference method for a system of abstract differential equations in a Hilbert space, which generalizes a number of linear systems of coupled thermoelasticity equations in the absence of any special conditions on the projection subspaces.     

Error estimates in the projection-difference method for a hyperbolic-parabolic system of abstract differential equations

We consider a Cauchy problem for a hyperbolic-parabolic system of abstract differential equations in a Hilbert space, which generalizes a number of linear coupled thermoelasticity problems. Energy error estimates for a projection-difference method as applied to the Cauchy problem are established without imposing any special conditions on projection subspaces.     

On the high smoothness of the solution of an abstract hyperbolic-parabolic system like systems of thermoelasticity equations

We study the Cauchy problem for a hyperbolic-parabolic system of abstract differential equations in a Hilbert space, which generalizes a number of linear coupled thermoelasticity problems. We establish results on the high smoothness of the solution with respect to time as well as with respect to the spatial variables under appropriate smoothness conditions on the right-hand side and data coordination conditions.     

Error estimate in the projection-difference method for an abstract quasilinear hyperbolic equation with a nonsmooth right-hand side

A three-level scheme of the projection-difference method for an abstract quasilinear hyperbolic equation is analyzed for convergency in a Hilbert space with variable operator coefficients and a non-smooth (only Bochner integrable) right-hand side. An energy error estimate is derived without any special conditions imposed on projection subspaces.     

Error estimates for schemes of the projection-difference method for abstract quasilinear hyperbolic equations

We study the convergence of the three-layer scheme of the projection-difference method for abstract quasilinear hyperbolic equations in Hilbert space. We establish asymptotic energy error estimates for an arbitrary choice of finite-dimensional subspaces in which the approximation problems are solved.     

On the stability of an operator-difference scheme of the type of hyperbolic-parabolic systems of differential equations

We obtain results on the stability of a three-layer operator-difference scheme that generalizes a class of difference and projection-difference schemes for linear coupled thermoelasticity problems.     

The convergence rate of a projection-difference method for an abstract coupled problem

We consider the Cauchy problem for a system of abstract differential equations in Hilbert spaces that generalizes some coupled thermoelasticity problems. We obtain an a priori energy estimate for the convergence rate of a projection-difference method as applied to the Cauchy problem with an arbitrary choice of projection subspaces.     

On a coupled system of abstract differential equations similar to the thermoelasticity equations

The study of a model system of differential equations arising from the dynamical problems of thermo elasticity is continued. The case of shifts of the general type is investigated. We employ the commutant method based on the properties of the operator (A,B)=AB-BA.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 8, pp. 1162–1165, August, 1993.     

Error estimate of a first-order time discretization scheme for the geodynamo equations

In this paper, we analyze a first-order time discretization scheme for a nonlinear geodynamo model and carry out the convergence analysis of this numerical scheme. It is concluded that our numerical scheme converges with first-order accuracy in the sense of L²$L^2$-norm with respect to the velocity field u$\mathit{u}$ and the magnetic field B$\mathit{B}$ and with half-order accuracy in time for the total kinematic pressure P.     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

Study of convergence of the projection-difference method for hyperbolic equations

We consider the Cauchy problem for an abstract quasilinear hyperbolic equation with variable operator coefficients and a nonsmooth but Bochner integrable free term in a Hilbert space. Under study is the scheme for approximate solution of this problem which is a combination of the Galerkin scheme in space variables and the three-layer difference scheme with time weights. We establish an a priori energy error estimate without any special conditions on the projection subspaces. We give a concrete form of this estimate in the case when discretization in the space variables is carried out by the finite element method (for a partial differential equation) and by the Galerkin method in Mikhlin form.     

Analysis of a hyperbolic-parabolic type system for image inpainting

In this paper we consider the initial boundary value problem of a hyperbolic-parabolic type system for image inpainting in a 2-D bounded domain, and establish the existence of weak solutions to the system by employing the method of vanishing viscosity.     

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

The present work considers a nonlinear abstract hyperbolic equation with a self-adjoint positive definite operator, which represents a generalization of the Kirchhoff string equation. A symmetric three-layer semi-discrete scheme is constructed for an approximate solution of a Cauchy problem for this equation. Value of the gradient in the nonlinear term of the scheme is taken at the middle point. It makes possible to find an approximate solution at each time step by inverting the linear operator. Local convergence of the constructed scheme is proved. Numerical calculations for different model problems are carried out using this scheme.     

On some second-order non-linear systems of a hyperbolic-parabolic type

We consider local solutions to the Cauchy problem for a class of non-linear hyperbolic-parabolic systems generalizing the systems of elasticity and thermoelasticity. Our main purpose is to relax the usual regularity requirements to include the nonclassical solutions into considerations.     

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrödinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.     

Convergence estimates of a projection-difference method for an operator-differential equation

This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given.     

Mean-square accuracy estimates of a projection-difference method for weakly solvable quasilinear parabolic equations

In a separable Hilbert space, we consider a weakly solvable quasilinear parabolic problem, which is solved by an approximate projection-difference method. With respect to time, we use a linear Euler method implicit in the leading part. We obtain coercive mean-square accuracy estimates for the approximate solutions. These estimates imply the convergence of the approximate method and provide the convergence rate accurate in the approximation order both in time and in space.     

Methods of the secant type for systems of equations with symmetric jacobian matrix

Symmetric methods (SS methods) of the secant type are proposed for systems of equations with symmetric Jacobian matrix. The SSI and SS2 methods generate sequences of symmetric matrices J and H which approximate the Jacobian matrix and inverse one, respectively. Rank-two quasi-Newton formulas for updating J and H are derived. The structure of the approximations J and H is better than the structure of the corresponding approximations in the traditional secant method because the SS methods take into account symmetry of the Jacobian matrix. Furthermore, the new methods retain the main properties of the traditional secant method, namely, J and H are consistent approximations to the Jacobian matrix; the SS methods converge superlinearly; the sequential (n + 1)-point SS methods have the R-order at least equal to the positive root of t_n+1-1=0.