review lecture 12-13 · 2.29 numerical fluid mechanics fall 2011 – lecture 14. review lecture 13,...
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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
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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
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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.”
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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
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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!
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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)
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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
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i-1
i-2 i-1 i+1 i+2
i+1i
i
Grid 2h
Grid h
Image by MIT OpenCourseWare.
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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)
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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!
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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)!
≈
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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
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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
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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)
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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
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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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