2.29 Numerical Fluid Mechanics Lecture 11 Slides...2.29 Numerical Fluid Mechanics PFJL Lecture 11,...
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2.29 Numerical Fluid Mechanics
Spring 2015 – L
2.29 Numerical Fluid Mechanics PFJL Lecture 11, 1
ecture 11
REVIEW Lecture 10:• Classification of Partial Differential Equations (PDEs) and
ionsexamples with finite difference discretizat
– Parabolic PDEs
– Elliptic PDEs
– Hyperbolic PDEs
• Error Types and Discretization Properties:
– Consistency:
– Truncation error:
– Error equation: (for linear systems)
– Stability: (for linear systems)
– Convergence:
ˆˆ( ) 0 , ( ) 0x ˆˆ( ) 0 , ( ) 0x ( ) 0 , ( ) 0 ( ) 0 , ( ) 0x x( ) 0 , ( ) 0x( ) 0 , ( ) 0 ( ) 0 , ( ) 0x( ) 0 , ( ) 0xx ( ) 0 , ( ) 0 ( ) 0 , ( ) 0( ) 0 , ( ) 0 ( ) 0 , ( ) 0x xx x( ) 0 , ( ) 0x( ) 0 , ( ) 0 ( ) 0 , ( ) 0x( ) 0 , ( ) 0( ) 0 , ( ) 0x( ) 0 , ( ) 0 ( ) 0 , ( ) 0x( ) 0 , ( ) 0( ) 0 , ( ) 0 ( ) 0 , ( ) 0( ) 0 , ( ) 0 ( ) 0 , ( ) 0 ( ) 0 , ( ) 0 ( ) 0 , ( ) 0
ˆ( ) ( ) 0 when 0x x ˆ( ) ( ) 0 when 0( ) ( ) 0 when 0x ( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0x x( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0xx ( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0x xx x( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0 ( ) ( ) 0 when 0( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0( ) ( ) 0 when 0x( ) ( ) 0 when 0 ( ) ( ) 0 when 0x( ) ( ) 0 when 0
ˆ( ) ( ) ( ) for 0px x O x x ˆ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0O x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x x x( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0x x x xx x x x x x x x( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0( ) ( ) ( ) for 0O x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0O x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x x x x x x x x x x x x x x x( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0 ( ) ( ) ( ) for 0x x( ) ( ) ( ) for 0
ˆˆ ˆ( ) ( ) ( )x x x ˆˆ ˆˆˆ ˆˆ( ) ( ) ( )x x x ( ) ( ) ( ) ( ) ( ) ( )x x x x x x( ) ( ) ( )x x x( ) ( ) ( ) ( ) ( ) ( )x x x( ) ( ) ( )x x x x x x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x x x x x x x x x x x x( ) ( ) ( )x x x( ) ( ) ( ) ( ) ( ) ( )x x x( ) ( ) ( ) ( ) ( ) ( )x x x( ) ( ) ( ) ( ) ( ) ( )x x x( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1ˆ Const.x
1ˆ Const.x
xx
1ˆ ( )px x O x
1ˆx xx xx xx xx x x x x x x x x xx x x xx x x xx x x xx x x x x x x x x x x xx x x x x x x xx x x x x x x xx x x x x x x xx x x x x x x x 1 1 1 1 1 1
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PFJL Lecture 11, 2Numerical Fluid Mechanics2.29
2.29 Numerical Fluid Mechanics
Spring 2015 – Lecture 11
REVIEW Lecture 10, Cont’d:• Classification of PDEs and examples
• 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:
– Taylor Tables or Method of Undetermined Coefficients (Polynomial Fitting)
• Simply a more systematic way to solve for coefficients ai
m s
i j i xmi rj
u a ux
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PFJL Lecture 11, 3Numerical Fluid Mechanics2.29
FINITE DIFFERENCES – Outline for Today
• Classification of Partial Differential Equations (PDEs) and examples with
finite difference discretizations (Elliptic, Parabolic and Hyperbolic PDEs)
• Error Types and Discretization Properties
– Consistency, Truncation error, Error equation, Stability, Convergence
• Finite Differences based on Taylor Series Expansions
– Higher Order Accuracy Differences, with Example
– Taylor Tables or Method of Undetermined Coefficients (Polynomial Fitting)
• Polynomial approximations
– Newton’s formulas
– Lagrange polynomial and un-equally spaced differences
– Hermite Polynomials and Compact/Pade’s Difference schemes
– Boundary conditions
– Un-Equally spaced differences
– Error Estimation: order of convergence, discretization error, Richardson’s
extrapolation, and iterative improvements using Roomberg’s algorithm
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PFJL Lecture 11, 4Numerical Fluid Mechanics2.29
References and Reading Assignments
• Chapter 23 on “Numerical Differentiation” and Chapter 18 on “Interpolation” of “Chapra and Canale, Numerical Methods for Engineers, 2006/2010/2014.”
• 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”
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PFJL Lecture 11, 5Numerical Fluid Mechanics2.29
Finite Differences using Polynomial approximations
Numerical Interpolation:
“Historical” Newton’s Iteration FormulaStandard triangular family of polynomials
Divided Differences: ci = ?
Newton’s Computational Scheme
2
3
0 0
x2
+
By recurrence:
First
divided
differences
Second
divided
differences
Newton’s formula allow easy recursive
computation of the coefficients of a polynomial
of order n that interpolates n+1 data point
Derivative of that polynomial can then be
expressed as a function of these n+1 data
points (in our case, unknown fct values)
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PFJL Lecture 11, 6Numerical Fluid Mechanics2.29
Divided Differences
with equidistant step size impliedEquidistant Sampling
Triangular Family of Polynomials
Equidistant Sampling
Finite Differences using Polynomial approximations
Equidistant Newton’s Interpolation
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PFJL Lecture 11, 7Numerical Fluid Mechanics2.29
Numerical Differentiation using Newton’s algorithm
for equidistant sampling: 1st Order
Triangular Family of Polynomials
Equidistant Sampling
First orderx
f(x)
h
n=1First Derivatives
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PFJL Lecture 11, 8Numerical Fluid Mechanics2.29
x
f(x)
h h
n=2
Second order
Second Derivatives
n=3 Central Difference
Forward Differencen=2
Central Difference
Forward Difference
2 22 2
Numerical Differentiation using Newton’s algorithm
for equidistant sampling: 2nd Order
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PFJL Lecture 11, 9Numerical Fluid Mechanics2.29
Finite Differences using Polynomial approximations
Numerical Interpolation: Lagrange Polynomials
(Reformulation of Newton’s polynomial)
x
f(x)1
k-3 k-2 k-1 k k+1 k+2
Difficult to program
Difficult to estimate errors
Divisions are expensive
Important for numerical integration
Nodal basis in FE
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PFJL Lecture 11, 10Numerical Fluid Mechanics2.29
• Use the values of the function and its derivative(s) at given points k– For example, for values of the function and of its first derivatives at pts k
• General form for implicit/explicit schemes (here focusing on
space)
– Generalizes the Lagrangian approach by using Hermitian interpolation
• Leads to the “Compact difference schemes” or “ Pade’ schemes ”
• Are implemented by the use of efficient banded solvers
• Derivatives are then also unknowns
Hermite Interpolation Polynomials and
Compact / Pade’ Difference Schemes
1 1
( ) ( ) ( )n m
k k kk k k
uu x a x u b xx
m qs
i i j i xmi r i pj i
ub a ux
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PFJL Lecture 11, 11Numerical Fluid Mechanics2.29
FINITE DIFFERENCES: Higher Order Accuracy
Taylor Tables for Pade’ schemes
u u u 1d e (au bu cu ?j 1 j j 1)x x x xj 1 j j 1
j
2
j 1
ux j
j 1
j 1
j
j+1
Image by MIT OpenCourseWare.
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PFJL Lecture 11, 12Numerical Fluid Mechanics2.29
Sum each column starting from left and force the sums to be zero by proper choice of a, b, c, etc:
1 1 1 0 0 01 0 1 1 1 1
1 3 0 3 1 11 0 1 2 2 04
1 0 1 3 3 01 0 1 4 4 0
ab
a b c d ecde
Truncation error is sum of the first column that does not vanish in the table, here 6th column (divided by Δx):
4 5
5120xj
x ux
FINITE DIFFERENCES: Higher Order Accuracy
Taylor Tables for Pade’ schemes, Cont’d
α αφx( (
i+1
φx( (
i+
φx( (
i-1+ β i+1φ i-1φ
∆2 x
i+2φ i-2φ∆4 x
γ+=
Image by MIT OpenCourseWare.
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PFJL Lecture 11, 13Numerical Fluid Mechanics2.29
Compact / Pade’ Difference Schemes: Examples
We can derive family of compact centered approximations for up to 6th order using:
Comments:
• Pade’ schemes use
fewer computational
nodes and thus are
more compact than CDS
• Can be advantageous
(more banded systems!)
α αφx( (
i+1
φx( (
i+
φx( (
i-1+ β i+1φ i-1φ
∆2 x
i+2φ i-2φ∆4 x
γ+=
Scheme Truncation error γα β
CDS-2
CDS-4
Pade-4
Pade-6
φ3∆x( (23! 3x
φ5∆x( (43 3! 5x
13
φ7∆x( (67! 7x
4
φ5∆x( (45! 5x
0 1 0
14
32 0
043
13
13
149
19
'
'
.
Image by MIT OpenCourseWare.
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PFJL Lecture 11, 14Numerical Fluid Mechanics2.29
Higher-Order Finite Difference Schemes
Considerations
• Retaining more terms in Taylor Series or in polynomial
approximations allows to obtain FD schemes of increased
order of accuracy
• However, higher-order approximations involve more nodes,
hence more complex system of equations to solve and more
complex treatment of boundary condition schemes
• Results shown for one variable still valid for mixed derivatives
• To approximate other terms that are not differentiated: reaction
terms, etc
– Values at the center node is normally all that is needed
– However, for strongly nonlinear terms, care is needed (see later)
• Boundary conditions must be discretized
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PFJL Lecture 11, 15Numerical Fluid Mechanics2.29
Finite Difference Schemes:
Implementation of Boundary conditions
• For unique solutions, information is needed at boundaries
• Generally, one is given either:
• 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
bnd
bnd bnd
bnd( , )
i) the variable: ( , ) ( ) (Dirichlet BCs)
ii) a gradient in a specific direction, e.g.: = (t) (Neumann BCs)
iii) a linear cx t
u x x t u tux
ombination of the two quantities (Robin BCs)
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PFJL Lecture 11, 16Numerical Fluid Mechanics2.29
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,
• 1st order forward-difference:
• Parabolic fit to the bnd point and two inner points:
• Cubic fit to 4 nodes (3rd order difference):
• Compact schemes, cubic fit to 4 pts:
• In Open-boundary systems, boundary problem is not well posed =>
– Separate treatment for inflow/outflow points, multi-scale (embedded) approach and/or generalized inverse problem (using data in the interior)
bnd
2 11 2
( , ) 2 1
0 0x t
u u u u ux x x
bnd
2 2 2 23 2 1 2 3 1 1 3 1 2 1 3 2 1
( , ) 2 1 3 1 3 2
( ) ( ) ( ) ( ) 4 3 for equidistant nodes( )( )( ) 2x t
u x x u x x u x x x xu u u ux x x x x x x x
bnd
34 3 2 1
( , )
2 9 18 11 ( ) for equidistant nodes6x t
u u u u u O xx x
2 3 4( , ) 1
1
18 9 2 6 for equidistant nodes11 11bndx t
u u u x uu ux
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PFJL Lecture 11, 17Numerical Fluid Mechanics2.29
2 3
1( 1)
( ) ( ) ( )( ) ( ) ( ) '( ) ''( ) '''( ) ... ( )2! 3! !
( ) ( )1!
nni i i
i i i i i i n
nni
n
x x x x x xf x f x x x f x f x f x f x Rn
x xR fn
• Truncation error depends not only on grid spacing but also on the
derivatives of the variable
• 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: Lets define and write the Taylor
series at :
Finite-Differences on Non-Uniform Grids: 1-D
2 3
1
1( 1)
( ) ( ) '( ) ''( ) '''( ) ... ( )2! 3! !
( )1!
nn
i i i i i i n
nn
n
x x xf x f x x f x f x f x f x Rn
xR fn
1 1 1,i i i i i ix x x x x x
ix
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PFJL Lecture 11, 18Numerical Fluid Mechanics2.29
• Evaluate f(x) at xi+1 and xi-1 , subtract results, lead to central-difference
• 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
– If the grid contracts/expands with a constant factor re :
– Leading truncation error term is :
– 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
2 31 1 1
1 1
2 3
1
( ) ( ) '( ) ''( ) '''( ) ... ( )2! 3! !
( )( ) ( ) '( ) ''( ) '''( ) ... ( )2! 3! !
nni i i
i i i i i i i n
nni i i
i i i i i i i n
x x xf x f x x f x f x f x f x Rn
x x xf x f x x f x f x f x f x Rn
2 2 3 31 1 1 1
1 1 1 1 1 1
( ) ( )'( ) ''( ) '''( ) ...2! ( ) 3! ( )
i i i i i ii i i n
i i i i i i
f x f x x x x xf x f x f x Rx x x x x x
x = Truncation error
1i e ix r x
(1 ) ''( )2
er e ix i
r x f x
1 1 1i i i ix x x x
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PFJL Lecture 11, 19Numerical Fluid Mechanics2.29
• What also matters is: “rate of error reduction as grid is refined”!
• 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:
which is since
– The factor R = 4 if re = 1 (uniform grid). R is actually minimum at re = 1.
– When re > 1 (expending grid) or re < 1 (contracting grid), the factor R > 4
Non-Uniform Grids Example: 1-D Central-difference
2,2
,
(1 )''( )
2(1 )
''( )2
he h i
i
he h i
i
r xf x
Rr x
f x
2 21 ,2h h
i e h ix r x
1 ,i e h ix r x
2,
,
(1 )e h
e h
rR
r
2
1 , 1( 1)hi i i e h ix x x r x
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PFJL Lecture 11, 20Numerical Fluid Mechanics2.29
Non-Uniform Grids Example: 1-D Central-difference
Conclusions
• When a non-uniform “geometric progression” 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 an area of active research in CFD
• Conclusions also valid for higher dimensions and for other methods
(finite elements, etc)
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2.29 Numerical Fluid MechanicsSpring 2015
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