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.

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.

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.

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.

scholarly journals Research of a third-order nonlinear differential equation in the vicinity of a moving singular point for a complex plane

2021 ◽  
Vol 263 ◽  
pp. 03019
Author(s):  
Victor Orlov ◽  
Magomedyusuf Gasanov
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Nonlinear Differential Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Third Order ◽  
Modified Method ◽  
Order Nonlinear Differential Equation ◽  
Priori Estimates ◽  
The Right

This article generalizes the previously obtained results of existence and uniqueness theorems for the solution of a third-order nonlinear differential equation in the vicinity of moving singular points in the complex domain, as well as constructs an analytical approximate solution, and obtains a priori estimates of the error of this approximate solution. The study was carried out using the modified method of majorants to solve this equation, which differs from the classical theory, in which this method is applied to the right-hand side of the equation The final point of the article is to conduct a numerical experiment to test the theoretical positions obtained.

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.

scholarly journals On estimates for the Fu–Yau generalization of a Strominger system

2019 ◽  
Vol 2019 (751) ◽  
pp. 243-274 ◽  
Author(s):  
Duong H. Phong ◽  
Sebastien Picard ◽  
Xiangwen Zhang
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Higher Dimensions ◽  
Hessian Equation ◽  
Right Hand ◽  
Priori Estimates ◽  
Dimension 2

AbstractWe study an equation proposed by Fu and Yau as a natural n-dimensional generalization of a Strominger system that they solved in dimension 2. It is a complex Hessian equation with right-hand side depending on gradients. Building on the methods of Fu and Yau, we obtain {C^{0}}, {C^{2}}, and {C^{2,\alpha}} a priori estimates. We also identify difficulties in extending the Fu–Yau arguments for non-degeneracy from dimension 2 to higher dimensions.

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.

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