On the validity of the modified equation approach to the stability analysis of finite-difference methods

1987 ◽  
Author(s):  
SIN-CHUNG CHANG
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Equation Approach ◽  
Difference Methods ◽  
The Stability ◽  
Modified Equation

scholarly journals Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Olufemi Bosede ◽  
Ashiribo Wusu ◽  
Moses Akanbi
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Stability And Convergence ◽  
Point Boundary ◽  
Difference Methods ◽  
The Stability

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

NUMERICAL INTERFACES IN FINITE DIFFERENCE METHODS FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

1993 ◽  
Vol 01 (02) ◽  
pp. 151-184 ◽  
Author(s):  
TAO LIN
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Discontinuous Coefficients ◽  
Interface Problems ◽  
Artificial Boundary ◽  
Numerical Examples ◽  
Difference Methods

In this paper, we discuss the interface problems arising in using finite difference methods to solve hyperbolic equations with discontinuous coefficients. The schemes developed here can be used to handle four important types of numerical interfaces due to: (1) the discontinuity of the coefficients of the PDE, (2) using artificial boundary, (3) using different finite difference formulae in different areas, and (4) using different grid sizes in different areas. Stability analysis for these schemes is carried out in terms of conventional l1, l2, and l∞ norms so that the convergence rates of these schemes are obtained. Several numerical examples are supplied to demonstrate properties of these schemes.

scholarly journals Numerical Solutions of the Modified Burgers’ Equation by Finite Difference Methods

2017 ◽  
Vol 13 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Yusuf Ucar ◽  
Nuri Murat Yagmurlu ◽  
Orkun Tasbozan
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Finite Difference Methods ◽  
Solution Process ◽  
Modified Burgers Equation ◽  
Finite Difference Equations ◽  
Difference Methods ◽  
Linearization Techniques

Abstract In this study, a numerical solution of the modified Burgers’ equation is obtained by the finite difference methods. For the solution process, two linearization techniques have been applied to get over the non-linear term existing in the equation. Then, some comparisons have been made between the obtained results and those available in the literature. Furthermore, the error norms L2 and L∞ are computed and found to be sufficiently small and compatible with others in the literature. The stability analysis of the linearized finite difference equations obtained by two different linearization techniques has been separately conducted via Fourier stability analysis method.

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.

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