scholarly journals Stability of the Difference Schemes for the Equations of Weakly Compressible Liquid

2007 ◽  
Vol 7 (3) ◽  
pp. 208-220 ◽  
Author(s):  
P. Matus ◽  
O. Korolyova ◽  
M. Chuiko
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Initial Conditions ◽  
Compressible Liquid ◽  
Stability Conditions ◽  
Differential Problem ◽  
Weakly Compressible ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Abstract A priory estimates of the stability in the sense of the initial data of the difference scheme approximating weakly compressible liquid equations in the Riemann invariants have been obtained. These estimates have been proved without any assumptions about the properties of the solution of the differential problem and depend only on the behavior of the initial conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for a finite instant of time t≤t_0. In particular, this is confirmed by the fact, that nonfulfilment of these stability conditions lead to the appearance of supersonic flows or domains with large gradients. The questions of uniqueness and convergence of the difference solution are considered also. The results of the computating experiment confirming the theoretical conclusions are given.

On The Stability Of Finite-Difference Schemes For Parabolic Equations Subject To Integral Conditions With Applications To Thermoelasticity

2008 ◽  
Vol 8 (4) ◽  
pp. 360-373 ◽  
Author(s):  
M SAPAGOVAS ◽  
Z. JESEVICIUTE
Keyword(s):  
Stability Analysis ◽  
Difference Scheme ◽  
Parabolic Equations ◽  
Finite Difference Schemes ◽  
Stability Conditions ◽  
Spectral Structure ◽  
Integral Conditions ◽  
Implicit Difference Scheme ◽  
The Difference ◽  
The Stability

Abstract The stability of implicit difference scheme for parabolic equations subject to integral conditions, which correspond to the quasi-static flexure of a thermoelastic rod is considered. The stability analysis is based on the spectral structure of matrix of the difference scheme. The stability conditions obtained here differ from those presented in the articles of other authors..

Nonlinear Stability Of The Difference Schemes For Equations Of Isentropic Gas Dynamics

2008 ◽  
Vol 8 (2) ◽  
pp. 155-170 ◽  
Author(s):  
P. MATUS ◽  
A. KOLODYNSKA
Keyword(s):  
Initial Data ◽  
Numerical Experiment ◽  
Difference Scheme ◽  
Nonlinear Stability ◽  
Gas Dynamics ◽  
Differential Problem ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas ◽  
The Difference ◽  
Theoretical Results

AbstractFor the difference scheme approximating the gas dynamics problem in Riemann invariants a priory estimates with respect to the initial data have been obtained. These estimates are proved without any assumptions about the solution of the differential problem using only limitations for the initial and boundary conditions. Estimates of stability in the general case have been obtained only for the finite instant of time. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment confirming theoretical results are given.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

Stochastic storage networks: stationarity and the feedforward case

1997 ◽  
Vol 34 (02) ◽  
pp. 498-507 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Total Variation ◽  
Finite Time ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Input Process ◽  
Stationary Increments ◽  
Stationary Version ◽  
Storage Network ◽  
The Difference ◽  
Storage Networks

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

scholarly journals Stability Analysis of the Explicit Difference Scheme for Richards Equation

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 352
Author(s):  
Fengnan Liu ◽  
Yasuhide Fukumoto ◽  
Xiaopeng Zhao
Keyword(s):  
Difference Scheme ◽  
Numerical Experiments ◽  
Richards Equation ◽  
Induction Method ◽  
Explicit Difference Scheme ◽  
Time Step ◽  
The Difference ◽  
The Stability ◽  
Forward Euler ◽  
Parabolic Term

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

scholarly journals A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Unique Solvability ◽  
Parabolic Type ◽  
Stability Estimates ◽  
The Difference ◽  
The Stability ◽  
Integral Differential Equation

The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

scholarly journals ANALYSIS OF STABILITY OF ONE CLASS OF NUMERICAL METHODS OF SECOND ORDER

2019 ◽  
pp. 92-95
Author(s):  
Vasyl Mykhaylovych Zaiats
Keyword(s):  
Numerical Methods ◽  
Second Order ◽  
Stability Conditions ◽  
Difference Formula ◽  
Conservative Systems ◽  
Trapezoid Method ◽  
The Difference ◽  
Analysis Of Stability ◽  
The Stability ◽  
Combined Numerical Methods

The paper deals with the analysis of the stability of a class of second order combined numerical methods based on the trapezoid method and the difference formula, what obtained by the author. The stability conditions for this class of methods are obtained by the example of conservative systems.

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