Download presentation

Presentation is loading. Please wait.

Published byBrianna Keene Modified over 3 years ago

1
Finite Difference Discretization of Elliptic Equations: 1D Problem Lectures 2 and 3

2
Model Problem Poisson Equation in 1D Boundary Value Problem (BVP) Describes many simple physical phenomena (e.g.): Deformation of an elastic bar Deformation of a string under tension Temperature distribution in a bar

3
Model Problem Poisson Equation in 1D Solution Properties The solution always exists is always smoother then the data If for all, then for all Given the solution is unique

4
Numerical Finite Differences Discretization Subdivide interval into equal subintervals for

5
Numerical Solution Finite Differences Approximation For example … forsmall

6
Numerical Solution Finite Differences Equations suggests …

7
Numerical Solution Finite Differences ….Equations (Symmetric)

8
Numerical Solution Finite Differences Solution Isnon-singular ? For any Hence for any ( Is APD) exists and is unique

9
Numerical Solution Finite Differences Example … with Take

10
Numerical Solution Finite Differences … Example

11
Numerical Solution Finite Differences Convergence ? 1. Does the discrete solution retain the qualitative propeties of the continuous solution ? 2. Does the solution become more accurate when ? 3. Can we make for arbierarily small ?

12
Discretization Error Analysis Properties of A -1 Let Non-negativity for Boundedness for

13
Discretization Error Analysis Qualitative Properties of If for Then for

14
Discretization Error Analysis Qualitative Properties of Discrete Stability

15
Discretization Error Analysis Truncation Error For any we can show that Take

16
Discretization Error Analysis Error Equation Let be the discretization error Subtracting and

17
Discretization Error Analysis Error Equation

18
Discretization Error Analysis Convergence Using the discrete stability estimate on or A-priori Error Estimate

19
Discretization Error Analysis Numerical Example

20
EXAMPLE : Asymptotically,

21
Discretization Error Analysis Summary For a simple model problem we can produce numerical approximations of arbitrary accuracy. An a-priori error estimate gives the asymptotic dependence of the solution error on the discretization size.

22
Generalizations Definitions Consider a linear elliptic differential equation and a difference scheme

23
Generalizations Consistency The difference scheme is consistent with the differential equation if: For all smooth functions for when for all is order of accuracy

24
Generalizations Truncation Error or, for The truncation error results from inserting the exact solution into the difference scheme. Consistency

25
Generalizations Error Equation Original scheme Consistercy The errorsatisfies

26
Generalizations Stability Matrix norm The difference scheme is stable if (independent of )

27
Generalizations Stability (max row sum)

28
Generalizations Convergence Error equation Taking norms

29
Generalizations Summary Consistency + Stability Convergence ConvergenceStability Consistency

30
The Eigenvalue Problem Model Problem Statement Find nontrivial such that denote solutions with

31
The Eigenvalue Problem Application Axially Loaded Beam Small Deflection External Force Equilibrium

32
The Eigenvalue Problem Exact Solution or

33
The Eigenvalue Problem Exact Solution Thus (choose ) Larger more oscillatory larger

34
The Eigenvalue Problem Exact Solution

35
The Eigenvalue Problem Discrete Equations Difference Formulas

36
The Eigenvalue Problem Discrete Equations Matrix Form

37
The Eigenvalue Problem Error Analysis Analytical Solution: Claim that Notesince

38
The Eigenvalue Problem Error Analysis Analytical Solution:

39
What are ?

40
The Eigenvalue Problem Error Analysis Analytical Solution: Thus:

41
The Eigenvalue Problem Error Analysis Conclusions … Low modes For fixed, : second-order convergence

42
The Eigenvalue Problem Error Analysis … Conclusions … High modes : For as High modes ( ) are not accurate.

43
The Eigenvalue Problem Error Analysis … Conclusions … Low modes vs. high modes Example :

44
The Eigenvalue Problem Error Analysis … Conclusions … Low modes vs. high modes resolved accurate not resolved not accurate is BUT: as,, so any fixed mode converges.

45
The Eigenvalue Problem Error Analysis … Conclusions …

47
The Eigenvalue Problem Condition Number of A For a SPD matrix, the condition number is given by = maximum eigenvalue of minimum eigenvalue of Thus, for our matrix, as grows (in ) as number of grid points squared. Importance: understanding solution procedures.

48
The Eigenvalue Problem Link to … Discretization … Recall : or

49
The Eigenvalue Problem Link to … Discretization … Error equation : for as(consistency)

50
The Eigenvalue Problem Link to Norm Discretization We will use the modified norm for Thus, from consistency

51
The Eigenvalue Problem Link to ||.|| Discretization … Ingredients: 1. Rayleigh Quotient : 2. Cauchy-Schwarz Inequality : for all

52
The Eigenvalue Problem Link to ||.|| Discretization … Convergence proof:

53
The Eigenvalue Problem Link to ||.|| Discretization …

54
Alternative Derivation Since From error equation Multiplying by

Similar presentations

OK

Computational Fluid Dynamics Lecture II Numerical Methods and Criteria for CFD Dr. Ugur GUVEN Professor of Aerospace Engineering.

Computational Fluid Dynamics Lecture II Numerical Methods and Criteria for CFD Dr. Ugur GUVEN Professor of Aerospace Engineering.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Incident command post ppt online Ppt on north bengal tourism Free ppt on arithmetic progression for class 10 Ppt on intelligent manufacturing journal Ppt on waxes lipids Ppt on direct and online marketing Ppt on non ferrous minerals definition Themes free download ppt on pollution Ppt on tsunami warning system Ppt on tamper resistant fasteners