scholarly journals Stability theory for some scalar finite difference schemes: validity of the modified equations approach

2021 ◽  
Vol 70 ◽  
pp. 124-136
Author(s):  
Firas Dhaouadi ◽  
Emilie Duval ◽  
Sergey Tkachenko ◽  
Jean-Paul Vila
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Finite Difference ◽  
Heat Equation ◽  
Infinite Series ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Modified Equations ◽  
The Stability ◽  
Modified Equation

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial differential equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.

scholarly journals Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method

2020 ◽  
pp. 8-19
Author(s):  
Mehwish Naz Rajput ◽  
Asif Ali Shaikh ◽  
Shakeel Ahmed Kamboh
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Stability Condition ◽  
Difference Method ◽  
Stable Solution ◽  
Finite Difference Schemes ◽  
Step Size ◽  
Stability Range ◽  
The Stability

Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain. Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain. Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.  Methodology: The stability condition over an elliptical domain with the non-uniform step size depending upon the boundary tracing function is derived by using Von Neumann method. Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution. Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.

scholarly journals Stability analysis of finite difference schemes for an advection diffusion equation

2017 ◽  
Vol 29 (2) ◽  
pp. 143-151 ◽  
Author(s):  
TMAK Azad ◽  
LS Andallah
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Time Step ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
The Stability ◽  
Stability Results

The paper studies stability analysis for two standard finite difference schemes FTBSCS (forward time backward space and centered space) and FTCS (forward time and centered space). One-dimensional advection diffusion equation is solved by using the schemes with appropriate initial and boundary conditions. Numerical experiments are performed to verify the stability results obtained in this study. It is found that FTCS scheme gives better point-wise solutions than FTBSCS in terms of time step selection.Bangladesh J. Sci. Res. 29(2): 143-151, December-2016

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.

Removing the stability limit of the time–space domain explicit finite difference schemes for acoustic modeling with stability condition-based spatial operators

Geophysics ◽  
2021 ◽  
pp. 1-82
Author(s):  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Grid Spacing ◽  
Time Step ◽  
Space Domain ◽  
Wavenumber Range ◽  
Time Space ◽  
Explicit Finite Difference ◽  
The Stability

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time–space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. Additionally, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis and numerical modeling demonstrate the accuracy and effectiveness of the proposed method.

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.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

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