Symbolic-Numerical Method for the Stability Analysis of Difference Schemes on the Basis of the Catastrophe Theory

1995 ◽  
Vol 116 (1) ◽  
pp. 26-38 ◽  
Author(s):  
E.V. Vorozhtsov ◽  
B.Yu. Scobelev ◽  
V.G. Ganzha
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Catastrophe Theory ◽  
Difference Schemes ◽  
The Stability

scholarly journals A General Numerical Method for Analyzing the Linear Stability of Stratified Parallel Shear Flows

2014 ◽  
Vol 31 (12) ◽  
pp. 2795-2808 ◽  
Author(s):  
Tim Rees ◽  
Adam Monahan
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Analytical Approximations ◽  
Onset Of Turbulence ◽  
Parallel Shear Flows ◽  
Critical Layers ◽  
The Stability ◽  
Stability Results

Abstract The stability analysis of stratified parallel shear flows is fundamental to investigations of the onset of turbulence in atmospheric and oceanic datasets. The stability analysis is performed by considering the behavior of small-amplitude waves, which is governed by the Taylor–Goldstein (TG) equation. The TG equation is a singular second-order eigenvalue problem, whose solutions, for all but the simplest background stratification and shear profiles, must be computed numerically. Accurate numerical solutions require that particular care be taken in the vicinity of critical layers resulting from the singular nature of the equation. Here a numerical method is presented for finding unstable modes of the TG equation, which calculates eigenvalues by combining numerical solutions with analytical approximations across critical layers. The accuracy of this method is assessed by comparison to the small number of stratification and shear profiles for which analytical solutions exist. New stability results from perturbations to some of these profiles are also obtained.

scholarly journals Failure Process and Stability Analysis of Rock Blocks in a Large Underground Excavation Based on a Numerical Method

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Shijie Chen ◽  
Ming Xiao ◽  
Juntao Chen
Keyword(s):  
Stability Analysis ◽  
Numerical Analysis ◽  
Numerical Method ◽  
Failure Process ◽  
Surrounding Rock ◽  
Underground Excavation ◽  
Analysis Method ◽  
Safety Factors ◽  
Geological Conditions ◽  
The Stability

A numerical analysis method for block failure is proposed that is based on continuum mechanics. First, a mesh model that includes marked blocks was established based on the grid-based block identification method. Then, expressions of the contact force under various contact states were derived based on the explicit contact force algorithm, and a contact simulation method between blocks and the surrounding rock was proposed. The safety factors of the blocks were calculated based on the strength reduction method. This numerical analysis method can simulate both the continuous deformation of the surrounding rock and the discontinuous failure processes of the blocks. A simple example of a sliding block was used to evaluate the accuracy and rationality of the numerical method. Finally, combined with a deep underground excavation project under complex geological conditions, the stability of the blocks and rock were analyzed. The results indicate that the key blocks are damaged after excavation, the potentially dangerous blocks loosen and undergo large deformations, and the cracks between the blocks and the rock gradually increase as the excavation proceeds. The safety factors of the blocks change during the excavation. The numerical results demonstrate the influence of the surrounding rock on the failure process and on the stability of the blocks, and an effective analysis method is provided for the stability analysis of blocks under complex geological conditions.

scholarly journals Numerical Method for the Stability Analysis of Ideal MHD Modes with a Wide Range of Toroidal Mode Numbers in Tokamaks

2007 ◽  
Vol 2 (0) ◽  
pp. 010-010 ◽  
Author(s):  
Nobuyuki AIBA ◽  
Shinji TOKUDA ◽  
Takaaki FUJITA ◽  
Takahisa OZEKI ◽  
Ming S. CHU ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Numerical Method ◽  
Toroidal Mode ◽  
Wide Range ◽  
Ideal Mhd ◽  
The Stability

scholarly journals Solving Volterra Integrodifferential Equations via Diagonally Implicit Multistep Block Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Nur Auni Baharum ◽  
Zanariah Abdul Majid ◽  
Norazak Senu
Keyword(s):  
Stability Analysis ◽  
Numerical Solution ◽  
Numerical Method ◽  
Numerical Computation ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Block Method ◽  
The Stability ◽  
Predictor Corrector

The performance of the numerical computation based on the diagonally implicit multistep block method for solving Volterra integrodifferential equations (VIDE) of the second kind has been analyzed. The numerical solutions of VIDE will be computed at two points concurrently using the proposed numerical method and executed in the predictor-corrector (PECE) mode. The strategy to obtain the numerical solution of an integral part is discussed and the stability analysis of the diagonally implicit multistep block method was investigated. Numerical results showed the competence of diagonally implicit multistep block method when solving Volterra integrodifferential equations compared to the existing methods.

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

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