2.29 Numerical Fluid Mechanics Lecture 12 Slides€¦ · · 2017-12-282.29 Numerical Fluid Mechanics PFJL Lecture 12, 1 ... b a u x Dt §·w ¨¸ ... –For the numerical sol.:

PFJL Lecture 12, 1Numerical Fluid Mechanics2.29

2.29 Numerical Fluid Mechanics

Spring 2015 – Lecture 12

REVIEW Lecture 11:

• Finite Differences based Polynomial approximations

– Obtain polynomial (in general un-equally spaced), then differentiate as needed

• Newton’s interpolating polynomial formulas

• Lagrange polynomial

• Hermite Polynomials and Compact/Pade’s Difference schemes

• Finite Difference: Boundary conditions

– Different approx. at and near the boundary => impacts global order of accuracy and linear system to be solved

Triangular Family of Polynomials

(case of Equidistant Sampling,

similar if not equidistant)

0 0,

( ) ( ) ( ) with ( )n n

jk k k

k j j k k j

x xf x L x f x L x

x x

(Reformulation of Newton’s polynomial)

m qs

i i j i xmi r i pj i

ub a ux

(Use the values of the function and

its derivative(s) at nodes)


2.29 Numerical Fluid Mechanics

Spring 2015 – Lecture 12

REVIEW Lecture 11:

• Finite Difference: Boundary conditions

– Different approx. at and near the boundary => impacts linear system to be solved

• Finite-Differences on Non-Uniform Grids and Uniform Errors: 1-D

– If non-uniform grid is refined, error due to the 1st order term decreases faster than that of 2nd order term

– Convergence becomes asymptotically 2nd order (1st order term cancels)

• Grid-Refinement and Error estimation

– Estimation of the order of convergence and of the discretization error

– Richardson’s extrapolation and Iterative improvements using Roomberg’salgorithm


FINITE DIFFERENCES – Outline for Today

• Finite-Differences on Non-Uniform Grids and Uniform Errors: 1-D

• Grid Refinement and Error Estimation

• Fourier Analysis and Error Analysis

– Differentiation, definition and smoothness of solution for ≠ order of spatial operators

• Stability

– Heuristic Method

– Energy Method

– Von Neumann Method (Introduction) : 1st order linear convection/wave eqn.

• Hyperbolic PDEs and Stability

– Example: 2nd order wave equation and waves on a string

• Effective numerical wave numbers and dispersion

– CFL condition:

• Definition

• Examples: 1st order linear convection/wave eqn., 2nd order wave eqn., other FD schemes

– Von Neumann examples: 1st order linear convection/wave eqn.

– Tables of schemes for 1st order linear convection/wave eqn.


References and Reading Assignments

• Lapidus and Pinder, 1982: Numerical solutions of PDEs in Science and Engineering. Section 4.5 on “Stability”.

• Chapter 3 on “Finite Difference Methods” of “J. H. Ferzigerand M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3rd edition, 2002”

• Chapter 3 on “Finite Difference Approximations” of “H. Lomax, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, 2003”

• Chapter 29 and 30 on “Finite Difference: Elliptic and Parabolic equations” of “Chapra and Canale, Numerical Methods for Engineers, 2014/2010/2006.”

Grid-Refinement and Error estimation

• We found that for a convergent scheme, the discretization error ε is ofthe form: (recall: )

where R is the remainder

• The degree of accuracy and discretization error can be estimated between solutions obtained on systematically refined/coarsened grids-True solution u can be expressed either as:

-Thus, the exponent p can be estimated:

(need to eliminate u and then need 2 eqns. to eliminate both Δx and p, hence u4Δx )

-The discretization error on the grid Δx can be estimated by:

-Good idea: estimate p to check code. Is it equal to what it is supposed to be?

-When solutions on several grids are available, an approximation of higher accuracy can be obtained from the remainder: Richardson Extrapolation!

2.29 Numerical Fluid Mechanics PFJL Lecture 12, 5

( )pO x R

2 ' (2 ) '

px

px

u u x Ru u x R

2 4

2

log log2x x

x x

u upu u

2

2 1x x

x pu u

ˆ ˆˆ, ( ) 0 , ( ) 0x (recall: ) ˆ ˆˆˆ ˆˆˆ ˆ(recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) x(recall: ) (recall: ) (recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) (recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) x(recall: ) (recall: ) x(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) (recall: ) xx(recall: ) x(recall: ) (recall: ) x(recall: ) (recall: ) (recall: ) (recall: ) (recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) x(recall: ) (recall: ) x(recall: ) (recall: ) x(recall: ) (recall: ) x(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) x(recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: ) (recall: ) , ( ) 0 , ( ) 0(recall: )


Richardson Extrapolation and Romberg Integration

Richardson Extrapolation: method to obtain a third improved estimate of an

integral based on two other estimates

Trapezoidal Rule:

h

(grid space)

I(h)

I

h1h2

Richardson Extrapolation:

Consider:

2

2

2

Example

Assume:

)

For two different grid space h1 and h2:

From two O(h2),

we get an O(h4)!

≈


Romberg’s Integration:Iterative application of Richardson’s extrapolation

h

I(h)

I

h1h2

Romberg Integration Algorithm, for any order k

1: O(h2) 2: O(h4) 3: O(h6) 4: O(h8)

a. 0.172800 1.367467

1.068800

b,. 0.172800 1.367467 1.640533

1.068800 1.623467

1.484800

c. 0.172800 1.367467 1.640533 1.640533

1.068800 1.623467 1.640533

1.484800 1.639467

1.600800

For Order 2 (case of previous slide):

j

k

Increasing

resolution

Increasing order


Romberg’s Differentiation:Iterative application of Richardson’s extrapolation

h

D(h)

D

h1h2

‘Romberg’ Differentiation Algorithm, for any order k

1: O(h2) 2: O(h4) 3: O(h6) 4: O(h8)

a. 0.172800 1.367467

1.068800

b,. 0.172800 1.367467 1.640533

1.068800 1.623467

1.484800

c. 0.172800 1.367467 1.640533 1.640533

1.068800 1.623467 1.640533

1.484800 1.639467

1.600800

j

kh3

D D

D

D

4D2,1 – D1,1

For Order 2 (as previous slide, but for differentiation):

Increasing

resolution

Increasing order


Fourier (Error) Analysis:

Definitions

• Leading error terms and discretization error estimates can be

complemented by a Fourier error analysis

• Fourier decomposition:

– Any arbitrary periodic function can be decomposed into its Fourier

components:

– Note: rate at which | fk | with |k| decays determine smoothness of f (x)• Examples drawn in lecture: sin(x), Gaussian exp(-πx 2), multi-frequency functions,

etc

2

02

0

( ) ( integer, wavenumber)

2 (orthogonality property)

1 ( )2

i kxk

k

ik x im xkm

ik xk

f x f e k

e e

f f x e dx

Using the orthog. property,

taking the integral/FT of f(x):



Differentiations

• Consider the decompositions:

• Taking spatial derivatives gives:

• Taking temporal derivatives gives:

• Hence, in particular, for even or odd spatial derivatives:

( ) or ( , ) ( )ikx ikxk k

k kf x f e f x t f t e

( )n

n ikxkn

k

f f t ik ex

( )r rikxk

r rk

f d f t et d t

2

2 1

2 ( 1) (real)

2 1 ( 1) (imaginary)

n q q

n q q

n q ik k

n q ik i k



Generic equation

• Consider the generic PDE:

• Fourier Analysis:

• Hence:

• Thus:

• And:

– “Phase speed”:

n

nf ft x

( ) ( ) nikx ikxk

kk k

d f t e f t ik edt

( ) ( ) ( ) for n nk

k kd f t ik f t f t ik

d t

( ) (0) , ( , ) (0)t ikx tk k k

kf t f e f x t f e

/c ik

( , ) ( ) i kx

k

k

f x t f t e



Generic equation

• Generic PDE, FT:

• Hence:

• Etc

22

2

233

3

44

4

Propagation: / 1,1

No dispersion

2 Decay

Propagation: / ,3

With dispersion

4 +: (Fast) Growth, : (Fast) Decay

c ikf fn ikt xf fn kt x

c ik kf fn ikt x

f fn kt x

2

2 1

2 ( 1) (real)

2 1 ( 1) (imaginary)

n q q

n q q

n q ik k

n q ik i k

( ) ( ) for nk

kd f t f t ik

d t

( , ) (0) i kx tk

kx t f e

f


Fourier Error Analysis: 1st derivatives

• In the decomposition:

– All components are of the form:

– Exact 1st order spatial derivative:

– However, if we apply the centered finite-difference (2nd order accurate):

– keff = effective wavenumber

– For low wavenumbers (smooth functions):

• Shows the 2nd order nature of center-difference approx. (here, of k by keff)

( ) ikxkf t e

( ) ( ) ( )

i kxikx ikxk

k kf t e f t ik e f t ik e

x

1 1

( ) ( )

eff

eff

2

sin( )2 2

sin( )where (uniform grid resolution )

jj jj j

j j

j

i kxi k x ik xik x x ik x xikxikx ikx

j

f ffx x

e e ee e e k xi e i k ex x x x

k xk xx

3 2

effsin( ) ...

6k x k xk kx

fx

( , ) ( ) i kx

k

k

f x t f t e


Fourier Error Analysis, Cont’d:

Effective Wave numbers

• Different approximations have different effective wavenumbers

– CDS, 2nd order:

– CDS, 4th order:

– Pade scheme, 4th order:

3 2

effsin( ) ...

6k x k xk kx

effsin( ) 4 cos( )

3k xk k x

x

eff3 sin( )

2 cos( )i k xi k

k x x

i kx

j

ex

kmax Δxmax

max

0 effk k

kx

Note that keff is bounded:

© Springer. All rights reserved. This content is excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/fairuse.Source: Lomax, H., T. Pulliam, D. Zingg. Fundamentals of ComputationalFluid Dynamics. Springer, 2001.

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Fourier Error Analysis, Cont’d

Effective Wave Speeds

Different approximations also lead to different effective wave speeds:

• Consider linear convection equations:

– For the exact solution:

– For the numerical sol.: if

which we can solve exactly (our interest here is only error due to spatial approx.)

– Often, ceff / c < 1 => numerical solution is too slow.

– Since ceff is a function of the effective wavenumber,

the scheme is dispersive (even though the PDE is not)

i kx

j

ex

0f fct x

( )( , ) (0) (0) (since )ikx t ik x ct

k kk k

f x t f e f e i k c

eff eff( )

eff eff eff

( , ) (0) (0)

(defining )

i k c t ik x cnumerical ikx tk k

k k

eff eff eff

f x t f e f e

c ki k c i k c

c k

.

. . .eff( ) ( ) ( )j j

num ikxikx ikxnum ikx num numk

k k kj

d f ef f t e e f t c f t c i k edt x

eff.( ) (0) ik c tnumk kf t f e

effcc

© Springer. All rights reserved. This content is excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/fairuse.Source: Lomax, H., T. Pulliam, D. Zingg. Fundamentals of ComputationalFluid Dynamics. Springer, 2001.

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Evaluation of the Stability of a FD Scheme:

Three main approachesRecall: Stability: (for linear systems)

• Heuristic stability:

– Stability is defined with reference to an error (e.g. round-off) made in

the calculation, which is damped (stability) or grows (instability)

– Heuristic Procedure: Try it out

• Introduce an isolated error and observe how the error behaves

• Requires an exhaustive search to ensure full stability, hence mainly

informational approach

• Energy Method

– Basic idea:

• Find a quantity, L2 norm e.g.

• Shows that it remains bounded for all n

– Less used than Von Neumann method, but can be applied to

nonlinear equations and to non-periodic BCs

• Von Neumann method (Fourier Analysis method)2.29 Numerical Fluid Mechanics PFJL Lecture 12, 16

2n

jj

ˆˆ ˆ( ) ( ) ( )x x x Recall: Stability: (for ˆˆ ˆˆˆ ˆˆ( ) ( ) ( )Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( ) ( ) ( ) ( )Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for xRecall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for Recall: Stability: (for ( ) ( ) ( )Recall: Stability: (for 1ˆ Const.x

Recall: Stability: (for 1Recall: Stability: (for 1Recall: Stability: (for ˆRecall: Stability: (for Const.Recall: Stability: (for xRecall: Stability: (for xRecall: Stability: (for Recall: Stability: (for Recall: Stability: (for

Recall: Stability: (for

Recall: Stability: (for xxRecall: Stability: (for xRecall: Stability: (for

Recall: Stability: (for xRecall: Stability: (for Recall: Stability: (for Recall: Stability: (for


Evaluation of the Stability of a FD Scheme

Energy Method Example

• Consider again:

• A possible FD formula (“upwind” scheme for c>0):

(t = nΔt, x = jΔx) which can be rewritten:

0ct x

11 0

n n n nj j j jc

t x

11(1 ) with n n n

j j jc t

x

jj-1n

n+1

x

t

Derivation removed due to copyright restrictions. For the rest of this derivation,please see equations 2.18 through 2.22 inDurran, D. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, 1998. ISBN: 9780387983769.


Evaluation of the Stability of a FD Scheme

Energy Method Example

Derivation removed due to copyright restrictions. For the rest of this derivation, please see equations 2.18 through 2.22 inDurran, D. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, 1998. ISBN: 9780387983769.


Von Neumann Stability

• Widely used procedure

• Assumes initial error can be represented as a Fourier Series and considers growth or decay of these errors

• In theoretical sense, applies only to periodic BC problems and to linear problems

– Superposition of Fourier modes can then be used

• Again, use, but for the error:

• Being interested in error growth/decay, consider only one mode:

• Strict Stability: The error will not to grow in time if

– in other words, for t = nΔt, the condition for strict stability can be written:

Norm of amplification factor ξ smaller or equal to1

( , ) ( ) i xx t t e

( ) where is in general complex and function of : ( )i x t i xt e e e

1te

1 or for , 1t te e von Neumann condition

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2.29 Numerical Fluid Mechanics Lecture 12 Slides€¦ · · 2017-12-282.29 Numerical Fluid Mechanics PFJL Lecture 12, 1 ... b a u x Dt §·w ¨¸ ... –For the numerical sol.: