REVIEW Lecture 12-13 · 2.29 Numerical Fluid Mechanics Fall 2011 – Lecture 14. REVIEW Lecture 13,...

2.29 Numerical Fluid MechanicsFall 2011 – Lecture 14

REVIEW Lecture 12-13: • Classification of PDEs and examples of finite-difference discretization

• Error Types and Discretization Properties • Finite Differences based on Taylor Series Expansions

– Higher Order Accuracy Differences, with Examples • Incorporate more higher-order terms of the Taylor series expansion than

strictly needed and express them as finite differences themselves (making them function of neighboring function values)

• If these finite-differences are of sufficient accuracy, this pushes the remainder to higher order terms => increased order of accuracy of the FD method

• General approximation: mu s

a u m i ji x x j ir

– Taylor Tables or Method of Undetermined Coefficients (Polynomial Fitting)• More systematic (Tables) way to solve for coefficients ai of higher-order FD

2.29 Numerical Fluid Mechanics PFJL Lecture 14, 1 1

2.29 Numerical Fluid MechanicsFall 2011 – Lecture 14

REVIEW Lecture 13, cont’d: • Finite Differences based Polynomial approximations

– Obtain polynomial (in general un-equally spaced), then differentiate if needed • Newton’s interpolating polynomial formulas

Triangular Family of Polynomials (case of Equidistant Sampling, similar if not equidistant)

• Lagrange polynomial n n x xf x x ( ) j

(Reformulation of Newton’s polynomial) ( ) Lk ( ) ( f xk ) with Lk x k 0 j0, jk xk x j

• Hermite Polynomials and Compact/Pade’s Difference schemes s m q (Use the values of the function and b u

a u i m i ji xits derivative(s) at nodes) ir x j i i p


FINITE DIFFERENCES – Outline for Today

• Polynomial approximations – Newton’s formulas – Lagrange polynomial and un-equally spaced differences – Hermite Polynomials and Compact/Pade’s Difference schemes

• Finite Difference: Boundary conditions • Finite-Differences on Non-Uniform Grids and Uniform Errors: 1-D • Grid Refinement and Error Estimation:

– Order of convergence, discretization error, Richardson’s extrapolation and Iterative improvements using Roomberg’s algorithm

• 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)

• Hyperbolic PDEs 2.29 Numerical Fluid Mechanics PFJL Lecture 14, 3

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References and Reading Assignments

• 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 ComputationalFluid Dynamics (Scientific Computation). Springer, 2003”

• Chapter 23 on “Numerical Differentiation” and Chapter 18 on “Interpolation” of “Chapra and Canale, Numerical Methods for Engineers, 2010/2006.”


Finite Difference Schemes: Implementation of Boundary conditions

• For unique solutions, information is needed at boundaries • Generally, one is given either:

i) the variable: u x( x , ) t u ( )t (Dirichlet BCs) bnd bnd

uii) a gradient in a specific direction, e.g.: = bnd (t) (Neumann BCs) x ( xbnd , ) t

iii) a linear c ombination of the two quantities (Robin BCs)

• Straightforward cases: – If value is known, nothing special needed (one doesn’t solve for the BC) – If derivatives are specified, for first-order schemes, this is also

straightforward to treat


Finite Difference Schemes: Implementation of Boundary conditions, Cont’d

• Harder cases: when higher-order approximations are used – At and near the boundary: nodes outside of domain would be needed

• Remedy: use different approximations at and near the boundary – Either, approximations of lower order are used – Or, approximations go deeper in the interior and are one-sided. For example,

u u2 u1• 1st order forward-difference: 0 0 u2 u1x t x2 x( xbnd , ) 1

• Parabolic fit to the bnd point and two inner points: 2 2 2 2u x( x ) u (x x ) u (x x ) (x x ) u 3 2 1 2 3 1 1 3 1 2 1 u3 4 u2 3u1 for equidistant nodes x , ) (x )( x x x2 )x x )( 2x ( xbnd t 2 1 3 1 3

u 2u 9u 18u 11u4 3 2 1 3• Cubic fit to 4 nodes (3rd order difference): ( ) for equidistant nodes O xx 6x( xbnd , ) t

18u 9u 2u 6x u 2 3 4• Compact schemes, cubic fit to 4 pts: u( t u1 for equidistant nodes , ) xbnd 11 11 x 1

• In Open-boundary systems, boundary problem is not well posed => –Separate treatment for inflow/outflow points, multi-scale approach and/or

generalized inverse problem (using data in the interior)2.29 Numerical Fluid Mechanics PFJL Lecture 14, 6

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Finite-Differences on Non-Uniform Grids: 1-D

• Truncation error depends not only on grid spacing but also on the derivatives of variable

2 3 nx x x nf (x ) f (x ) x f '(x ) f ''(x ) f '''(x ) ... f (x ) Ri1 i i i i i n2! 3! n! n1x ( 1) R f n ( )n n 1!

• Uniform error distribution can not be achieved on a uniform grid => non-uniform grids – Use smaller (larger) Δx in regions where derivatives of the function are

large (small) => uniform discretization error

– However, in some approximation (centered-differences), specific terms cancel only when the spacing is uniform

• Example: let’s define x x x , x x xi1 i1 i i i i1

2 3 n(x x ) (x x ) (x x ) ni i if ( ) f (x ) (x x ) f '(x ) f ''(x ) f '''(x ) ... f (x x ) Ri i i i i i n2! 3! n!and n1(x xi ) ( 1) nR f ( )n n 1!


Non-Uniform Grids Example: 1-D Central-difference

• Evaluate f(x) at xi+1 and xi-1 , subtract results, lead to central-difference 2 3 nx x x ni1 i1 i1f (x ) f (x ) x f '(x ) f ''(x ) f '''(x ) ... f (x ) Ri1 i i1 i i i i n2! 3! n!

2 3 nx x ( x )i i i n( ) f x ) x f '(x ) f ''(x ) f '''(x ) ... f (f x ( x ) Ri1 i i i i i i n2! 3! n! 2 2 3 3( ) f x ) x x xf x ( xi1 i1 i1 i i1 if '(x ) f ''(x ) f '''(x ) ... Ri i i nx x 2! ( x x ) 3! ( x x )i1 i1 i1 i1 i1 i1

= Truncation error x

• For a non-uniform mesh, the leading truncation term is O(Δx)

– The more non-uniform the mesh, the larger the 1st term in truncation error

x– If the grid contracts/expands with a constant factor re : xi1 re i

(1 r ) xre e i– Leading truncation error term is : x f ''(xi )2 – If re is close to one, the first-order truncation error remains small: this is

good for handling any types of unknown function f(x)


Non-Uniform Grids Example: 1-D Central-difference

• However, what matters is: “rate of error reduction as grid is refined”!

x 2h r x2hi1 e h, i 2

ix 1 r , xe i h

• Consider case where refinement is done by adding more grid points but keeping a constant ratio of spacing (geometric progression), i.e.

• For coarse grid pts to be collocated with fine-grid pts: (re,h)2 = re,2h

• The ratio of the two truncation errors at a common point is then: (1 r ,2 e h i f ''(x ) (1 2

2 i rR which is R , e h ) since x2h

(1 r ) xh r i x i xi 1 (r , xi1 1 e h ) ,e h i f ''(x , e h

i )2 – The factor R = 4 if re = 1 (uniform grid)

– When re > 1 (expending grid) or re < 1 (contracting grid), the factor R > 4

) x2h


i-1

i-2 i-1 i+1 i+2

i+1i

i

Grid 2h

Grid h

Image by MIT OpenCourseWare.

Non-Uniform Grids Example: 1-D Central-differenceConclusions

• When a non-uniform grid is refined, error due to the 1st order term decreases faster than that of 2nd order term !

• Since (re,h)2 = re,2h , we have re,h → 1 as the grid is refined. Hence, convergence becomes asymptotically 2nd order (1st order term cancels)

• Non-uniform grids are thus useful, if one can reduce Δx in regions where derivatives of the unknown solution are large • Automated means of adapting the grid to the solution (as it evolves)

• However, automated grid adaptation schemes are more challenging in higher dimensions and for multivariate (e.g. physics-biology-acoustics) or multiscale problems

• (Adaptive) Grid generation still a very challenging problem in CFD

• Conclusions also valid for higher dimensions and for other methods (finite elements, etc)


Grid-Refinement and Error estimation

• We found that for a convergent scheme, the discretization error ε is of ˆ ˆ ˆ O x p ) ( ) 0 )

where R is the remainder

the form: ( R (recall: , L ( ) 0 , Lx

• This discretization error can be estimated between solutions obtained on systematically refined/coarsened grids

pu ux x R -True solution u can be expressed either as:

u u ' (2 x) p R ' 2x

-Thus, the exponent p can be estimated: 2 4

2

log log 2 x x

x x

u u p u u

(need two equations to eliminate both Δx and p, hence u4Δx )

u ux 2x-The discretization error on the grid Δx can be estimated by: x p2 1

-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!


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

Romberg Integration Algorithm, for any order k

h

I(h)

I

For Order 2 (case of previous slide):

h2 h1

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

a. 0.172800 1.068800

b. 0.172800 1.068800 1.484800

c. 0.172800 1.367467 1.640533 1.640533j 1.068800 1.623467 1.640533

1.484800 1.639467 1.600800

1.367467

1.367467 1.640533 1.623467


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

‘Romberg’ Differentiation Algorithm, for any order k

h3 h2 h1

k

h

D(h)

D

D D

D

D

4D2,1 – D1,1

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

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

a. 0.172800 1.068800

b,. 0.172800 1.068800 1.484800

c. 0.172800 1.367467 1.640533 1.640533j 1.068800 1.623467 1.640533

1.484800 1.639467 1.600800

1.367467

1.367467 1.640533 1.623467


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:

( ) fk eikx (f x k integer, wavenumber) k

2

i k x im x e e 2km (orthogonality property) 0

1 2Using the orthog. property, i k x x e dx taking the integral/FT of f(x): f f ( )k 2 0

– 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 2.29 Numerical Fluid Mechanics PFJL Lecture 14, 15

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Fourier (Error) Analysis: Differentiations

• Consider the decompositions:

i kx i kx ( ) fk e or f x t fk ( ) t ef x ( , ) k k

n f n i kx • Taking spatial derivatives gives:n fk ( )t ik e

x k

r r k i kx • Taking temporal derivatives gives: f

d f ( )t e

t r k d t r

• Hence, in particular, for even or odd spatial derivatives:q 2q2q n k (real) n ik ( 1)

q 2 1n 2q 1 n i ( 1) qik k (imaginary)


Fourier (Error) Analysis: Generic equation

f f• Consider the generic PDE: n

t xn

x t ( ) t eikx • Fourier Analysis: f ( , ) fk k

d fk ( )t nikx i kx fk ( ) ik • Hence: e t e

k d t k

d fk ( )t n n• Thus: f ( ) fk t for ik ik k t ( ) d t

t i kx t• And: f t( ) fk (0) e , f (x t , ) f (0) ek k

k

– “Phase speed”: c / ik


Fourier (Error) Analysis: Generic equation

d fk ( )t n n• Generic PDE, FT: ik fk ( )t fk ( )t for ik d t

q 2q2q n k (real) n ik ( 1) q 2 1n n ( 1) q2q 1 ik i k (imaginary)

• Hence: f f Propagation: c / ik 1,

n 1 ik t x No dispersion

f 2 f 2n 2 k Decay t x2

f 3 f Propagation: c / ik k 2 ,n 3 ik 3

t x3 With dispersion f 4 f n 4 4 k 4 +: (Fast) Growth, : (Fast) Decayt x

• Etc


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2.29 Numerical Fluid Mechanics Fall 2011

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REVIEW Lecture 12-13 · 2.29 Numerical Fluid Mechanics Fall 2011 – Lecture 14. REVIEW Lecture 13,...

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