scholarly journals COEFFICIENT STABILITY OF OPERATOR–DIFFERENCE SCHEMES

1999 ◽  
Vol 4 (1) ◽  
pp. 135-146
Author(s):  
P. P. Matus ◽  
B. S. Jovanović
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Evolutionary Equation ◽  
Finite Difference Schemes ◽  
Initial Condition ◽  
First Order ◽  
Time Integral ◽  
Priori Estimates ◽  
Coefficient Stability

A priori estimates expressing continuous dependence of the solution of a first order evolutionary equation in Hubert space on initial condition, right hand side and operator perturbations are obtained in time–integral norms. Analogous results hold for corresponding finite difference schemes.

On the Strong Stability of Operator-Difference Schemes in Time-Integral Norms

2001 ◽  
Vol 1 (1) ◽  
pp. 72-85 ◽  
Author(s):  
Boško S. Jovanović ◽  
Piotr P. Matus
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Stability ◽  
Difference Schemes ◽  
Initial Condition ◽  
Time Integral ◽  
Right Hand ◽  
Priori Estimates ◽  
The Stability

Abstract In this paper we investigate the stability of two-level operator-difference schemes in Hilbert spaces under perturbations of operators, the initial condition and right hand side of the equation. A priori estimates of the error are obtained in time- integral norms under some natural assumptions on the perturbations of the operators.

The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation

2021 ◽  
Vol 24 (4) ◽  
pp. 1231-1256
Author(s):  
Anatoly Alikhanov ◽  
Murat Beshtokov ◽  
Mani Mehra
Keyword(s):  
A Priori Estimates ◽  
Theoretical Calculations ◽  
A Priori ◽  
Approximation Error ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Test Problems ◽  
Priori Estimates ◽  
Uniform Grids ◽  
Crank Nicolson

Abstract In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracy O ( h 4 + τ 2 − α ) $\mathcal{O}(h^4+\tau^{2-\alpha})$ and O ( h 4 + τ 2 ) $\mathcal{O}(h^4+\tau^{2})$ . A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems.

scholarly journals FIRST-ORDER METHODS FOR GENERALIZED OPTIMAL CONTROL PROBLEMS FOR SYSTEMS WITH DISTRIBUTED PARAMETERS

2020 ◽  
pp. 18-44
Author(s):  
S. V. Denisov ◽  
V. V. Semenov
Keyword(s):  
Optimal Control Problems ◽  
A Priori Estimates ◽  
A Priori ◽  
Control Problems ◽  
First Order ◽  
Admissible Sets ◽  
Distributed Parameters ◽  
Priori Estimates ◽  
Systems With Distributed Parameters ◽  
First Order Methods

The problems of optimization of linear distributed systems with generalized control and first-order methods for their solution are considered. The main focus is on proving the convergence of methods. It is assumed that the operator describing the model satisfies a priori estimates in negative norms. For control problems with convex and preconvex admissible sets, the convergence of several first-order algorithms with errors in iterative subproblems is proved.

Monotone and Economical Difference Schemes on Nonuniform Grids for a Multidimensional Parabolic Equation with Third Kind

2004 ◽  
Vol 4 (3) ◽  
pp. 350-367
Author(s):  
Piotr Matus ◽  
Grigorii Martsynkevich
Keyword(s):  
Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Nonuniform Grids ◽  
The Third ◽  
Priori Estimates ◽  
The Difference

AbstractMonotone economical difference schemes of the second order of local approximation with respect to space variables on nonuniform grids for the heat con- duction equation with the boundary conditions of the third kind in a p-dimensional parallelepiped are constructed. The a priori estimates of stability and convergence of the difference solution in the norm C are obtained by means of the grid maximum principle.

INVARIANT DIFFERENCE SCHEMES FOR PARABOLIC EQUATIONS WITH TRANSFORMATIONS OF INDEPENDENT VARIABLES

1999 ◽  
Vol 09 (01) ◽  
pp. 93-110 ◽  
Author(s):  
A. A. SAMARSKII ◽  
V. I. MAZHUKIN ◽  
P. P. MATUS ◽  
V. G. RYCHAGOV ◽  
I. SMUROV
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
Order Approximation ◽  
A Priori ◽  
Parabolic Type ◽  
Difference Schemes ◽  
Invariance Property ◽  
Independent Variables ◽  
Priori Estimates ◽  
Nonstationary Equations

In this paper, invariant difference schemes for nonstationary equations under independent variables transformation constructed and investigated. Under invariance of difference scheme we mean its ability to preserve basic properties (stability, approximation, convergency, etc.) in various coordinate systems. Difference schemes of the second-order approximation that satisfy the invariance property are constructed for equations of parabolic type. Stability and convergency investigation of correspondent difference problems are carried out; a priori estimates in various grid norms are obtained.

STABILITY OF THREE‐LEVEL DIFFERENCE SCHEMES WITH RESPECT TO THE RIGHT‐HAND SIDE

2004 ◽  
Vol 9 (3) ◽  
pp. 243-252
Author(s):  
E. L. Zyuzina
Keyword(s):  
Wave Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Order Of Approximation ◽  
Level Difference ◽  
Right Hand ◽  
Priori Estimates ◽  
Uniform Grids ◽  
The Right

In this paper we investigate three‐level difference schemes on non‐uniform grids in time. The a priori estimates of stability with respect to the initial data and the right‐hand side are obtained. New schemes of the raised order of approximation for wave equations are constructed and investigated.

Asymptotic Stability of a Three-level Operator-difference Scheme

2006 ◽  
Vol 6 (4) ◽  
pp. 405-412
Author(s):  
Boško S. Jovanović
Keyword(s):  
Asymptotic Stability ◽  
Difference Scheme ◽  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Priori Estimates

Abstract Asymptotic stability of linear three-level operator-difference schemes is investigated in the case of commutative operators. Some new a priori estimates are obtained.

Difference Schemes for Elliptic Equations with Mixed Derivatives

2004 ◽  
Vol 4 (4) ◽  
pp. 494-505 ◽  
Author(s):  
Piotr Matus ◽  
Irina Rybak
Keyword(s):  
Differential Equation ◽  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Priori Estimates ◽  
Conservative Difference Schemes ◽  
Mixed Derivatives ◽  
Conservative Difference

AbstractIn this paper, the a priori estimates of stability in the energy and the uniform norms are proved for the monotone and conservative difference schemes approximating elliptic equations with mixed derivatives. The estimates are obtained without any assumption about the symmetry of the coe±cient matrix of the initial differential equation.

Three-Level Difference Schemes on Non-Uniform in Time Grids

2001 ◽  
Vol 1 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Piotr Matus ◽  
Elena Zyuzina
Keyword(s):  
Hilbert Spaces ◽  
Wave Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Nonuniform Grids ◽  
Priori Estimates ◽  
Long Time ◽  
The Right

Abstract In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

STABILITY AND CONVERGENCE OF TWO-LEVEL DIFFERENCE SCHEMES IN INTEGRAL WITH RESPECT TO TIME NORMS

1998 ◽  
Vol 08 (06) ◽  
pp. 1055-1070 ◽  
Author(s):  
ALEXANDER A. SAMARSKII ◽  
PETR P. MATUS ◽  
PETR N. VABISHCHEVICH
Keyword(s):  
Hilbert Spaces ◽  
A Priori Estimates ◽  
A Priori ◽  
Mathematical Physics ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Priori Estimates ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Nowadays the general theory of operator-difference schemes with operators acting in Hilbert spaces has been created for investigating the stability of the difference schemes that approximate linear problems of mathematical physics. In most cases a priori estimates which are uniform with respect to the t norms are usually considered. In the investigation of accuracy for evolutionary problems, special attention should be given to estimation of the difference solution in grid analogs of integral with respect to the time norms. In this paper a priori estimates in such norms have been obtained for two-level operator-difference schemes. Use of that estimates is illustrated by convergence investigation for schemes with weights for parabolic equation with the solution belonging to [Formula: see text].

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