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.

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].

Stability of Difference Schemes for Nonlinear Time-dependent Problems

2003 ◽  
Vol 3 (2) ◽  
pp. 313-329 ◽  
Author(s):  
Piter Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Quasilinear Parabolic Equations ◽  
Basic Point ◽  
Nonlinear Part ◽  
Priori Estimates ◽  
The Difference ◽  
The Stability

AbstractIn the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.

scholarly journals SOME ESTIMATES FOR A SPECIAL LINEAR DIFFERENCE PARABOLIC EQUATION

1997 ◽  
Vol 2 (1) ◽  
pp. 152-159
Author(s):  
Artūras Štikonas
Keyword(s):  
Parabolic Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
A Priori Estimates ◽  
A Priori ◽  
Linear Parabolic Equation ◽  
Special Linear ◽  
Priori Estimates ◽  
The Difference

The finite‐difference scheme for a special linear parabolic equation is investigated. A priori estimates for such finite‐difference scheme are derived in the difference analogues of norm on Banach function spaces V2 and W 2 2,1.

On the Stability of Differential-operator Equations and Operator-difference Schemes as t → ∞

2002 ◽  
Vol 2 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Boško Jovanović ◽  
Sergey Lemeshevsky ◽  
Peter Matus
Keyword(s):  
Asymptotic Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Nonlinear Problems ◽  
Abstract Cauchy Problem ◽  
Difference Schemes ◽  
Operator Equations ◽  
Priori Estimates ◽  
The Difference ◽  
The Stability

AbstractFor the abstract Cauchy problem for a parabolic equation a priori estimates of the global and asymptotic stability in various energy norms have been obtained. Similar problems are also considered for the second-order equation. In the latter case, a priori estimates of the asymptotic stability by the initial data have been obtained. The corresponding estimates of the global stability for three-level operator difference schemes have been proved. Estimates of the asymptotic behavior of the solution for quasi-linear multidimensional equations with unbounded nonlinearity have been obtained. The corresponding mathematical apparatus permitting one to prove unconditional monotonicity of the difference schemes approximating nonlinear problems is presented.

scholarly journals MONOTONE ECONOMICAL SCHEMES FOR QUASILINEAR PARABOLIC EQUATIONS

2002 ◽  
Vol 7 (2) ◽  
pp. 207-216
Author(s):  
N. V. Dzenisenko ◽  
A. P. Matus ◽  
P. P. Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Quasilinear Parabolic Equation ◽  
Uniform Norm ◽  
Additive Scheme ◽  
Quasilinear Parabolic Equations ◽  
The Maximum Principle ◽  
Priori Estimates ◽  
The Difference ◽  
Quasilinear Parabolic

In order to approximate a multidimensional quasilinear parabolic equation with unlimited nonlinearity the economical vector‐additive scheme is constructed. It is shown that its solution satisfies the maximum principle and, hence, the scheme is monotone. The proof is based on the equivalence of the vector‐additive scheme and the scheme of summarized approximation (locally one‐dimensional scheme). The a priori estimates of the difference solution in the uniform norm are obtained.

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

scholarly journals Analytical solution of a class of nonlinear differential equations of the third order in the complex area

2021 ◽  
pp. 75-84
Author(s):  
Виктор Николаевич Орлов ◽  
Людмила Витальевна Мустафина
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
A Priori Estimates ◽  
Nonlinear Differential Equations ◽  
A Priori ◽  
A Priori Estimate ◽  
Third Order ◽  
The Third ◽  
Change Of Variables ◽  
Priori Estimates

В работе приводится доказательство теоремы существования и единственности аналитического решения класса нелинейных дифференциальных уравнений третьего порядка, правая часть которого представлена полиномом шестой степени, в комплексной области. Расширен класс рассматриваемых уравнений за счет новой замены переменных. Получена априорная оценка аналитического приближенного решения. Представлен вариант численного эксперимента оптимизации априорных оценок с помощью апостериорных. The article presents a proof of the theorem of the existence and uniqueness of the analytical solution of the class of nonlinear differential equations of the third order, with a polynomial right-hand side of the sixth degree, in the complex domain. The class of the considered equations has been extended by means of a new change of variables. An a priori estimate of the analytical approximate solution is obtained. A variant of the numerical experiment of optimizing a priori estimates using a posteriori estimates is presented.

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.

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