Stability and Oscillation Characteristics of Finite-Element, Finite-Difference, and Weighted-Residuals Methods for Transient Two-dimensional Heat Conduction in Solids

1973 ◽  
Vol 95 (2) ◽  
pp. 235-239 ◽  
Author(s):  
R. V. S. Yalamanchili ◽  
S.-C. Chu
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Von Neumann ◽  
Weighted Residuals ◽  
Difference Formula ◽  
The Difference ◽  
The Stability ◽  
The Galerkin Method ◽  
Difference Expression ◽  
Oscillation Characteristics

The finite-element difference expression was derived by use of the variational principle and finite-element synthesis. Several ordinary finite-difference formulae for the La-placian term were considered. A particular finite-difference formula for the Laplacian term was chosen to bring the difference expressions of finite-element, finite-difference, and weighted-residuals (Galerkin) methods into the same format. The stability criteria were established for all three techniques by use of the general stability, von Neumann, and Dusinberre concepts. The oscillation characteristics were derived for all three techniques. The finite-element method is more conservative than the finite-difference method, but not so conservative as the Galerkin method in both stability and oscillation characteristics.

The Definition of a Critical Deflection of Compressed Rods with the Creep by the Method of Bubnov-Galerkin

2018 ◽  
Vol 931 ◽  
pp. 127-132
Author(s):  
Batyr M. Yazyev ◽  
Serdar B. Yazyev ◽  
Anatoly P. Grinev ◽  
Elena A. Britikova
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Methods ◽  
Difference Method ◽  
Stability Problem ◽  
The Finite Element Method ◽  
The Difference ◽  
The Stability ◽  
Definition Of ◽  
The Galerkin Method

The comparison of the numerical methods: the finite element method, the Galerkin Method, the difference method is considered for the study of the stability of the rods. The dependence of the solution of the stability problem on the parameters of the discretization of these numerical methods is studied. It is shown that the mathematical models are sufficiently accurate to analyze the stability of the rods of constant and variable sections.

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 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.

On the Stability of Multi-Step Finite-Difference-Based Lattice Boltzmann Schemes

2018 ◽  
Vol 16 (01) ◽  
pp. 1850087 ◽  
Author(s):  
Gerasim V. Krivovichev ◽  
Sergey A. Mikheev
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann ◽  
Time Derivative ◽  
Kinetic Equations ◽  
High Order ◽  
Numerical Characteristic ◽  
Central Difference ◽  
Boltzmann Equations ◽  
Von Neumann ◽  
The Stability

Stability of finite-difference-based off-lattice Boltzmann schemes is analyzed. The time derivative in system of discrete Boltzmann equations is approximated by two-step modified central difference. Advective term is approximated by finite differences from first- to fourth-orders of accuracy. Characteristics-based (CB) schemes and schemes with traditional separate approximations of space derivatives are considered. A special class of high-order CB schemes with approximation in the internal nodes of grid patterns is constructed. It is demonstrated that apparent viscosity for the schemes of high-order is equal to kinematic viscosity of the system of Bhatnaghar–Gross–Krook kinetic equations. Stability of the schemes is analyzed by the von Neumann method for the cases of two flow regimes in unbounded domain. Stability is analyzed by the investigation of the stability domains in parameter space. The area of the domain is considered as the main numerical characteristic of the stability. As the main result of the analysis, it must be mentioned that the areas of CB schemes are greater than areas for the schemes with separate approximations.

A recipe for stability of finite‐difference wave‐equation computations

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 967-969 ◽  
Author(s):  
Larry R. Lines ◽  
Raphael Slawinski ◽  
R. Phillip Bording
Keyword(s):  
Inverse Problem ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Numerical Stability ◽  
Short Note ◽  
Seismic Wave Propagation ◽  
Difference Wave ◽  
Von Neumann ◽  
The Stability

Finite‐difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation (Kelly and Marfurt, 1990) and in seismic imaging (Bording and Lines, 1997). That is, they are useful for the forward problem (modeling) and the inverse problem (migration). In computational solutions to the wave equation, it is necessary to be aware of conditions for numerical stability. In this short note, we examine a convenient recipe for insuring stability in our finite‐difference solutions to the wave equation. The stability analysis for finite‐difference solutions of partial differential equations is handled using a method originally developed by Von Neumann and described by Press et al. (1986, p. 827–830).

Finite Element Simulation and Stability Analysis of Composite Soil Nailing

2014 ◽  
Vol 580-583 ◽  
pp. 83-88 ◽  
Author(s):  
Jie Zhao ◽  
Ling Li Wang
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Soil Nailing ◽  
Deep Excavation ◽  
Soil Nail ◽  
Element Simulation ◽  
Normal Soil ◽  
Composite Soil Nailing ◽  
The Difference ◽  
The Stability

Under the condition of plane strain, a 2D elastoplastic FEM is used to analyze the behavior of composite soil nailing bracing of deep excavation, then finite element method of stability analysis is applied to evaluate the stability of the soil-nail wall. The authors analyzed the difference between composite soil nailing and normal soil nailing. Through analyzing the effect of bracing parameters on the deformation behavior and stability of the excavation, some useful conclusions are obtained to provide certain references for the design and construction of composite soil nailing.

A stability checking procedure for finite-difference schemes with boundary conditions in acoustic media

1989 ◽  
Vol 79 (5) ◽  
pp. 1601-1606
Author(s):  
Aladin H. Kamel
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Stability Condition ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Neumann Condition ◽  
Time Behavior ◽  
Von Neumann ◽  
The Stability ◽  
Stability And Accuracy

Abstract The manner in which boundary conditions are approximated and introduced into finite-difference schemes has an important influence on the stability and accuracy of the results. The standard von Neumann stability condition applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. In this paper, we extend the von Neumann condition to include boundary conditions. The method is based on studying the time propagating matrix which governs the space-time behavior of the numerical grid. Examples of applying the procedure on schemes with different boundary conditions are given.

scholarly journals Stability analysis of the implicit finite-difference-based upwind lattice Boltzmann schemes

2019 ◽  
pp. 116-127
Author(s):  
Г.В. Кривовичев ◽  
М.П. Мащинская
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Lattice Boltzmann ◽  
Kinetic Equations ◽  
Finite Difference Schemes ◽  
Courant Number ◽  
Von Neumann ◽  
Upwind Schemes ◽  
Implicit Finite Difference ◽  
The Stability

Статья посвящена анализу устойчивости неявных конечно-разностных схем для системы кинетических уравнений, применяемых для проведения гидродинамических расчетов в рамках метода решеточных уравнений Больцмана. Представлены семейства двухслойных и трехслойных схем с направленными разностями первого-четвертого порядков аппроксимации по пространственным переменным. Важной особенностью схем является то, что конвективные слагаемые аппроксимируются одной конечной разностью. Показано, что в выражении для аппроксимационной вязкости схем высоких порядков отсутствуют фиктивные слагаемые, что позволяет применять их во всем диапазоне значений времени релаксации. Анализ устойчивости проводится по линейному приближению с использованием метода Неймана. Получены приближенные условия устойчивости в виде неравенств на значения параметра Куранта. При расчетах показано, что площади областей устойчивости в пространстве параметров у двухслойных схем больше, чем у трехслойных. Исследованные схемы могут применяться при расчетах как непосредственно, так и в методах типа предиктор-корректор. The paper is devoted to the stability analysis of the implicit finite-difference schemes for the system of kinetic equations used for the hydrodynamic computations in the framework of the lattice Boltzmann method. The families of two- and three-layer upwind schemes of the first to fourth approximation orders on spatial variables are considered. An important feature of the presented schemes is that the convective terms are approximated by one finite difference. It is shown that, for the high-order schemes, in the expression for the current viscosity there are no fictitious terms, which makes it possible to perform computations in the whole range of relaxation time values. The stability analysis is based on the application of the von Neumann method to the linear approximations of the schemes. The stability conditions are obtained in the form of inequalities imposed on the Courant number values. It is also shown that the areas of stability domains for the two-layer schemes are greater than for the three-layer schemes in the parameter space. The considered schemes can be used as the fully implicit schemes in computational algorithms directly or in the predictor-corrector methods.

A Stability Analysis of Constrained Rotating Disks With Different Boundary Conditions

2007 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Boundary Conditions ◽  
High Speed ◽  
Equations Of Motion ◽  
Order Partial Differential Equation ◽  
Rotating Disks ◽  
Different Boundary Conditions ◽  
The Difference ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

The vibration behavior of constrained high speed rotating disks is of interest in industries as diverse as: aerospace, computer disk manufacture and saw design and usage. The purpose of this study is to investigate the stability behavior of guided circular disks with different boundary conditions. The equations of motion are developed for circular rotating disks constrained by space fixed linear, mass, spring, damper systems. The resulting equation of motion is a two dimensional fourth order partial differential equation that requires numerical solution. The Galerkin Method is employed using the eigenfunctions of the stationary non-constrained disk as approximation functions. Of interest is the effect on stability of conditions at the inner boundary. In particular the difference in behavior for centrally clamped, and splined disks (those disks that run on a spline arbor) is investigated. Also discussed is the effect of constraints on the flutter and divergence instability boundaries. Preliminary experimental results are presented for constrained splined disks, and these results are compared with the analytical predictions.

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