CFD-FDM_10.pdf
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The document discusses consistency, stability, and convergence for finite difference approximations of partial differential equations (PDEs). It states that a finite difference approximation is consistent if the difference between the PDE and approximation goes to zero as the grid is refined. It is stable if errors do not grow from one time step to the next. Von Neumann stability analysis determines stability by examining the growth of Fourier components of the numerical solution.
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![Consistency, Stability and Convergence
L.K.SAHA 97
truncation error for this finite-difference representation of the
heat equation is defined as the difference between the partial
differential equation and the difference approxi-mation to it.
That is, T.E. = PDE - FDE.
The order of the truncation error in this case is O(t) + O[(x)2
] which is frequently expressed in the form O[t, (x)2 ].
Naturally, we solve only the finite-difference equations and
hope that the truncation error is small.
How do we know that our difference representation is
acceptable and that a marching solution technique will work in
the sense of giving us an approximate solution to the PDE?
In order to be acceptable, our difference representation for this
marching problem needs to meet the conditions of consistency
and stability.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Fundamentals of Finite Difference Methods
This document provides an overview of finite difference methods for solving partial differential equations. It introduces partial differential equations and various discretization methods including finite difference methods. It covers the basics of finite difference methods including Taylor series expansions, finite difference quotients, truncation error, explicit and implicit methods like the Crank-Nicolson method. It also discusses consistency, stability, and convergence of finite difference schemes. Finally, it applies these concepts to fluid flow equations and discusses conservative and transportive properties of finite difference formulations.
poster2
The document compares several nonlinear and linear stabilization schemes (SUPG, dCG91, Entropy Viscosity) for solving advection-diffusion equations using finite element methods. It presents results of applying the different schemes to stationary and non-stationary test equations, comparing maximum overshoot and undershoot, smearing, and convergence orders. For both linear and quadratic elements, the nonlinear dCG91 and Entropy Viscosity schemes showed smaller overshoots and undershoots than linear schemes like SUPG and no stabilization.
Intro
This document provides an introduction to the basic concepts of computational fluid dynamics (CFD). It discusses the need for CFD due to the inability to analytically solve the governing equations for most engineering problems. The document then summarizes some common applications of CFD in industry, including simulating vehicle aerodynamics, mixing manifolds, and bio-medical flows. It also outlines the overall strategy of CFD in discretizing the continuous problem domain into a discrete grid before discussing specific discretization methods like the finite difference and finite volume methods.
Ch 05 MATLAB Applications in Chemical Engineering_陳奇中教授教學投影片
The slides of Chapter 5 of the book entitled "MATLAB Applications in Chemical Engineering": Numerical Solution of Partial Differential Equations. Author: Prof. Chyi-Tsong Chen (陳奇中教授); Center for General Education, National Quemoy University; Kinmen, Taiwan; E-mail: chyitsongchen@gmail.com.
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Solution of nonlinear_equations
This document provides an overview of solving nonlinear equations. It discusses solving single nonlinear equations and systems of nonlinear equations. It covers existence and uniqueness of solutions, multiple roots, sensitivity and conditioning. Iterative methods are used to solve nonlinear equations numerically. The bisection method is introduced, which successively halves the interval containing the solution until the desired accuracy is reached. The convergence rate of the bisection method is linear, meaning the error bound is reduced by half each iteration.
FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADI...
This document presents a finite element formulation using Finite Calculus (FIC) for solving convective-diffusive problems with sharp gradients. FIC modifies the governing equations by including characteristic length distances, which helps stabilize solutions. The FIC method is implemented by solving the modified governing equations using the Galerkin finite element method. Characteristic length vectors are computed based on principal curvature directions of the solution. Numerical examples demonstrate that FIC provides more accurate solutions than standard SUPG by better resolving sharp gradients.
Frequency Response Techniques
1. The document discusses frequency response analysis techniques, which analyze how a system responds to input signals of varying frequencies.
2. It describes two common frequency response techniques - Bode plots, which show magnitude and phase response as functions of frequency, and Nyquist plots, which plot magnitude against phase on a polar graph.
3. The techniques provide insights into system stability and dynamics and are useful for control system design, but their use requires complex derivations and they do not always directly indicate transient response characteristics.
Numerical_PDE_Paper
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
A Numerical Method For Friction Problems With Multiple Contacts
This document summarizes a numerical method for solving friction problems involving multiple contact surfaces. It begins by reviewing previous work on solving differential equations with discontinuities. The author then describes extending their previous method to handle problems with multiple contacts. Indicator functions are used to represent the regions of contact. Linear complementarity problems (LCPs) are solved to determine changes in the active contact surfaces. The method assumes the indicator functions and vector fields are smooth. Convergence results are proven showing the method can achieve high-order accuracy.
Project Paper
This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides examples of time series data and introduces the AR(1) model. The document describes an algorithm for calculating a bootstrap confidence interval for forecasting from an AR(1) model. It then discusses a simulation study comparing empirical coverage rates of bootstrap confidence intervals under different parameters. Finally, it applies the bootstrap method to forecasting Gross National Product growth, comparing the results to a parametric approach.
fb69b412-97cb-4e8d-8a28-574c09557d35-160618025920
This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides background on time series models and the autoregressive (AR) process. It then presents an algorithm for calculating a bootstrap confidence interval for forecasts from an AR(1) model. A simulation study compares coverage rates for bootstrap confidence intervals under different parameters. Finally, the method is applied to US Gross National Product data to forecast and construct confidence intervals.
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- 1. Consistency, Stability and Convergence L.K.SAHA 96 Truncation Error Consider the heat equation Using a forward-difference representation for the time derivative and a central difference representation for the space derivative, and include the truncation errors associated with the difference representation of the derivatives we obtain
- 2. Consistency, Stability and Convergence L.K.SAHA 97 truncation error for this finite-difference representation of the heat equation is defined as the difference between the partial differential equation and the difference approxi-mation to it. That is, T.E. = PDE - FDE. The order of the truncation error in this case is O(t) + O[(x)2 ] which is frequently expressed in the form O[t, (x)2 ]. Naturally, we solve only the finite-difference equations and hope that the truncation error is small. How do we know that our difference representation is acceptable and that a marching solution technique will work in the sense of giving us an approximate solution to the PDE? In order to be acceptable, our difference representation for this marching problem needs to meet the conditions of consistency and stability.
- 3. Consistency, Stability and Convergence L.K.SAHA 98 A finite-difference representation of a PDE is said to be consistent if we can show that the difference between the PDE and its difference representation vanishes as the mesh is refined, This should always be the case if the order of the truncation error vanishes under the grid refinement [i.e., O(t), O(x), etc.]. 0 0 . . lim lim . 0 mesh mesh i e PDE FDE T E
- 4. Consistency, Stability and Convergence L.K.SAHA 99 A finite difference approximation of a PDE is consistent if the finite difference equation approaches the PDE as the grid size approaches zero. Given a partial differential equation Pu = f and a finite difference scheme, Pt, x v = f, we say that the finite difference scheme is consistent with the partial differential equation if for any smooth function (x, t)
- 5. Consistency, Stability and Convergence L.K.SAHA 100
- 6. Consistency, Stability and Convergence L.K.SAHA 102 A stable numerical scheme is one for which errors from any source (round-off, truncation, mistakes) are not permitted to grow in the sequence of numerical procedures as the calculation proceeds from one marching step to the next. Generally, concern over stability occupies much more of our time and energy than does concern over consistency. Consistency is relatively easy to check and most schemes which are conceived will be consistent just due to the methodology employed in their development. Stability is much more subtle and usually a bit of hard work is required in order to establish analytically that a scheme is stable.
- 7. Von Neumann Stability Analysis L.K.SAHA 103 Von Neumann stability analysis is a commonly used procedure for determining the stability requirements of finite difference equations. In this method, a solution of the finite difference equation is expanded in a Fourier series. The decay or growth of the amplification factor indicates whether or not the numerical algorithm is stable. Recall that, for a linear equation, various solutions may be added. Therefore, when the FDE under investigation is linear, it is sufficient to investigate only one component of the Fourier series. In fact, the linearity of the equation is a general requirement for the application of the von Neumann stability analysis. Furthermore, the effect of the boundary condition on the stability of the solution is not included with this procedure.
- 8. Von Neumann Stability Analysis L.K.SAHA 104 To overcome these limitations, one may locally linearize the nonlinear equation and subsequently apply the von Neumann stability analysis. However, note that the resulting stability requirement is satisfied locally. Therefore, the actual stability requirement may be more restrictive than the one obtained from the von Neumann stability analysis. Nevertheless, the results will provide very useful information on stability requirements. To illustrate the procedure, assume a Fourier component for ui n, as