Projection-difference Schemes Research Articles

A projection-difference scheme with economical operator for the Stokes evolution equation in the cylinder with a small hole

A projection-difference scheme with economical operator for the Stokes evolution equation in the cylinder with a small hole

Stability of an operator-difference scheme for thermoelasticity problems

We study a linear three-layer operator-difference scheme with weights which generalizes a class of difference and projection-difference schemes for coupled thermoelasticity problems. Using the method of energy inequalities, we obtain stability estimates in grid energy norms under certain conditions on operator coefficients and parameters of the scheme.

Projection–difference method for controlled Fourier filtering

A projection---difference method is developed for approximating controlled Fourier filtering for quasilinear parabolic functional-differential equations. The method relies on a projection---difference scheme (PDS) for the approximation of the differential problem and derives a O(?1/2?+?h) bound on the rate of convergence of PDS in the weighted energy norm without prior assumptions of additional smoothness of the generalized solutions. The PDS leads to a natural approximation of the objective functional in the optimal Fourier filtering problem. A bound of the same order is obtained for the rate of convergence in the functional of the problems approximating the Fourier filter control problem.

Stability of a three-layer operator-difference scheme for coupled thermoelasticity problems

Stability of a three-layer operator-difference scheme with weights, which generalizes a class of difference and projection-difference schemes for linear coupled thermoelasticity problems, is analyzed. Energy estimates for the solution and its first-order grid derivative are obtained.

Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation

A novel technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a quasilinear parabolic operator-differential equation with initial data u0 ∈ H. The technique is based on the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the nonlinear term, a new estimate of order \(O(\sqrt \tau + h)\) for the convergence rate in a weighted energy norm is obtained without using a priori assumptions on the additional smoothness of weak solutions.

Weighted estimate for the convergence rate of a projection difference scheme for a parabolic equation and its application to the approximation of the initial-data control problem

A new technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a linear parabolic operator-differential equation. The technique is based on discrete analogues of weighted estimates reflecting the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the right-hand side, a new convergence rate estimate of order O(\( \sqrt \tau \) + h) is obtained in a weighted energy norm without making any a priori assumptions on the additional smoothness of weak solutions. The technique leads to a natural projection difference approximation of the problem of controlling nonsmooth initial data. The convergence rate estimate obtained for the approximating control problems is of the same order O(\( \sqrt \tau \) + h) as for the projection difference scheme.

Convergence rate estimates for a projection-difference scheme as applied to the nonstationary stokes equation in cylindrical coordinates

An implicit projection-difference scheme is constructed for the nonstationary Stokes equation in cylindrical coordinates. No axial symmetry is assumed. Under minimal assumptions about the initial data, convergence rate estimates are obtained that are uniform in the inner radius of the domain of order (τ1/2 + h)α, α = 1, 2. The results remain valid for domains with no hole and in the case of Cartesian coordinates.

Error estimates for projection-difference schemes for degenerate nonstationary equations

Error estimates for projection-difference schemes for degenerate nonstationary equations

Convergence of the Projection Difference Scheme for the Nonlinear Parabolic Equation with Transformed Spatial Argument

We consider a mixed initial boundary-value problem for the nonlinear parabolic functional-differential equation. The functional part of the equation depends on a generalized superposition of the sought solution and a transformation of the one-dimensional spatial argument. An approximate projection difference scheme is proposed for a wide class of measurable (including non-invertible) transformations. An O(τ + h1 + γ) bound is obtained for the rate of convergence in the L2(Q) norm to the generalized solutions of the original problem without prior assumptions about invertibility of the transformation, smoothness of the solution, or compatibility of grid increments.

Rate of convergence bound in the W 2 1 norm of the projection-difference scheme for the first boundary-value problem of elasticity theory in the anisotropic case

Exact difference scheme operators are applied to construct a projection-difference scheme for the system of equilibrium equations of a nonhomogeneous anisotropic elastic solid rigidly clamped in the rectangular region Ω. The scheme has O(h2) convergence in the W 12 (Ω) norm if the solution\(\overline u \)(x) is contained in the Sobolev space W 32 (Ω).

Difference methods in problems of the optimal control of the steady selfaction of light beams

A discussion is presented of optimization problems arising in the simulation of the propagation of optical radiation in a material medium with different regimes of absorption and field selfaction. Estimates are established for the rate of convergence of difference and projection-difference schemes for the appropriate non-linear differential equations, no prior assumptions being made as to the smoothness of the generalized solutions or compatability conditions for the grid step-sizes. Estimates are also obtained for the rate of convergence of the finite-dimensional approximations to the optimal control problems based on these schemes.

Exact estimates of the error gradient for projection-difference schemes for parabolic equations in an arbitrary domain

Estimates are given for the rate of convergence of some two-layer economic projection-difference schemes (p.d.s.) and Crank-Nicholson p.d.s. for solving parabolic equations in an arbitrary domain.

Estimates of the rate of convergence of projection-difference schemes for parabolic equations in a domain with a small opening

Projection-difference schemes are considered for linear parabolic equations in a cylindrical domain with a small hole of Diameter E. Estimates of the rate of convergence, uniform with respect to ε, weighted norms, are obtained in the classes of unsmooth right-hand sides and initial functions.

Difference schemes of projection type for symmetric systems of first-order equations

Projection difference schemes for symmetric positive first-order systems, based on multilinear mesh functions, are studied. It is shown that the schemes have accuracy of order h 1/2 . The efficiency of a two-step iterative method for solving the difference equations is proved.

The rate of convergence of the projection-difference scheme with a splitting operator for parabolic equations

The rate of convergence of the projection-difference scheme with a splitting operator for parabolic equations

Explicit and implicit projection-difference schemes for the solution of the Navier-Stokes equations

Explicit and implicit projection-difference schemes for the solution of the Navier-Stokes equations

Convergence rate estimate in L2 of projection-difference schemes for parabolic equations

PROJECTION-DIFFERENCE schemes for linear parabolic equations are considered. Oder-wise accurate convergence rate estimates in L 2 are obtained for several classes of right-hand sides and initial functions.