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PFJL Lecture 10, 1Numerical Fluid Mechanics2.29

2.29 Numerical Fluid Mechanics

Spring 2015 – Lecture 10

REVIEW Lecture 9:

• End of (Linear) Algebraic Systems

– Gradient Methods

– Krylov Subspace Methods

– Preconditioning of Ax=b

• FINITE DIFFERENCES

– Classification of Partial Differential Equations (PDEs) and examples

with finite difference discretizations

• Parabolic PDEs

• Elliptic PDEs

• Hyperbolic PDEs


FINITE DIFFERENCES - Outline

• Classification of Partial Differential Equations (PDEs) and examples with

finite difference discretizations

– Parabolic PDEs, Elliptic PDEs 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 approximations

– Newton’s formulas

– Lagrange polynomial and un-equally spaced differences

– Hermite Polynomials and Compact/Pade’s Difference schemes

– Equally spaced differences

• Richardson extrapolation (or uniformly reduced spacing)

• Iterative improvements using Roomberg’s algorithm


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”


Partial Differential Equations

Hyperbolic PDE: B2 - 4 A C > 0

Wave equation, 2nd orderExamples:

• Allows non-smooth solutions

• Information travels along characteristics, e.g.:

– For (3) above:

– For (4), along streamlines:

• Domain of dependence of u(x,T) = “characteristic path”

• e.g., for (3), it is: xc(t) for 0< t < T

• Finite Differences, Finite Volumes and Finite Elements

• Upwind schemes

2 22

2 2(1) u uct x

Unsteady (linearized) inviscid convection

(Wave equation first order)(3) ( )

t

uU u g

(4) ( ) U u gSteady (linearized) inviscid convection

t

x, y0

( ( ))d tdt

cc

xU x

dds

cxU

(2) 0u uct x

Sommerfeld Wave/radiation equation,

1st order



Hyperbolic PDE - ExampleWaves on a String

Initial Conditions

Boundary Conditions

Wave Solutions

t

xu(x,0), ut(x,0)

u(0,t) u(L,t)

Typically Initial Value Problems in Time, Boundary Value Problems in Space

Time-Marching Solutions:

Implicit schemes generally stable

Explicit sometimes stable under certain conditions

2 22

2 2

( , ) ( , ) 0 , 0u x t u x tc x L tt x


t

xu(x,0), ut(x,0)

u(0,t) u(L,t)


Hyperbolic PDE - Example

j+1

j-1j

ii-1 i+1

Finite Difference Representations (centered)

Finite Difference Representations

Wave Equation

2 22

2 2

( , ) ( , ) 0 , 0u x t u x tc x L tt x

Discretization:


t

xu(x,0), ut(x,0)

u(0,t) u(L,t)j+1

j-1j

ii-1 i+1

Introduce Dimensionless Wave Speed

Explicit Finite Difference Scheme

Stability Requirement:


Hyperbolic PDE - Example

1 Courant-Friedrichs-Lewy condition (CFL condition) c tCx

Physical wave speed must be smaller than the largest numerical wave speed, or,

Time-step must be less than the time for the wave to travel to adjacent grid points:

or x xc tt c

Error Types and Discretization Properties:

C

2.29 Numerical Fluid Mechanics PFJL Lecture 10, 8

onsistency

Consider the differential equation ( symbolic operator)

and its discretization for any given difference scheme

Consistency (Property of the discretization)

– The discretization of a PDE should asymptote to the PDE itself as the mesh-size/time-step goes to zero, i.e

for all smooth functions :

(the truncation error vanishes as mesh-size/time-step goes to zero)

( ) 0 ( ) 0( ) 0( ) 0( ) 0( ) 0

ˆˆ ( ) 0x ˆˆ ( ) 0x ( ) 0( ) 0xx ( ) 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

(


T


runcation error and Error equation

Truncation error

– Since , the truncation error is the result of inserting the exact

solution in the difference scheme

– If the FD scheme is consistent:

– p (>0) is the order of accuracy for the FD scheme

– Order p indicates how fast the error is reduced when the grid is refined

Error evolution equation

– From and where ε is the discretization error, for

linear problems, we have:

– The truncation error acts as a source for the discretization error, which is

convected, diffused, evolved, etc., by the operator

Remember:does not satisfy the FD eqn.

ˆxˆ

xxx

( ) 0 Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact Since , the truncation error is the result of inserting the exact Since , the truncation error is the result of inserting the exact ( ) 0( ) 0Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact Since , the truncation error is the result of inserting the exact ( ) 0Since , the truncation error is the result of inserting the exact

ˆˆ ( ) 0x From and where

linear problems, we have:

ˆˆ ( ) 0From and where ( ) 0From and where ( ) 0From and where ( ) 0From and where xFrom and where xFrom and where ( ) 0( ) 0From and where ( ) 0From and where From and where ( ) 0From and where From and where From and where xxFrom and where xFrom and where From and where xFrom and where From and where ( ) 0From and where From and where ( ) 0From and where

ˆ( ) ( )x x ˆ( ) ( )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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ˆ

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( ) ( ) ( )ˆ ( )

x x 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( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ˆ ( )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 x( ) ( )x x( ) ( )x x( ) ( )x x( ) ( )x x( ) ( )x x( ) ( )x x( )

ˆx

The truncation error acts as a source for the discretization error, which is ˆ

xxx


S


tability

Stability

– A numerical solution scheme is said to be stable if it does not amplify

errors ε that appear in the course of the numerical solution process

– For linear(-ized) problems, since , stability implies:

with the Const. not a function of

• If inverse was not bounded, discretization errors ε would increase with

iterations

– In practice, infinite norm is often used.

– However, difficult to assess stability in real cases due to boundary

conditions and non-linearities

• It is common to investigate stability for linear problems, with constant

coefficients and without boundary conditions

• A widely used approach: von Neumann’s method (see lectures 12-13)

x1ˆ Const.x

1ˆ Const.x

xx

ˆ ( )x x ) problems, since , ˆ) problems, since , ( )) problems, since , x x) problems, since , ) problems, since , ) problems, since , ( )) problems, since , ) problems, since , ( )) problems, since ,

) problems, since ,

) problems, since , ) problems, since , ( )) problems, since ,

) problems, since , ( )) problems, since , x x x x) problems, since , x x) problems, since ,

) problems, since , x x) problems, since , ( )x x( ) ( )x x( )) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since ,

) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since , ) problems, since , ) problems, since ,

) problems, since , ) problems, since , ) problems, since , ( )) problems, since , ) problems, since , ( )) problems, since ,

) problems, since , ( )) problems, since , ) problems, since , ( )) problems, since , ) problems, since , x x) problems, since , ) problems, since , x x) problems, since ,

) problems, since , x x) problems, since , ) problems, since , x x) problems, since , ) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since , ) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since ,

) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since , ) problems, since , ( )) problems, since , x x) problems, since , ( )) problems, since , ) problems, since , ) problems, since , ) problems, since , ) problems, since ,

x

1ˆ Const.x

infinite norm is often used.

1infinite norm is often used.

1infinite norm is often used.ˆ Const.infinite norm is often used.Const.infinite norm is often used.xinfinite norm is often used.xinfinite norm is often used.infinite norm is often used.

infinite norm is often used.infinite norm is often used.infinite norm is often used.xxinfinite norm is often used.xinfinite norm is often used.infinite norm is often used.xinfinite norm is often used.

infinite norm is often used.infinite norm is often used.


Convergence

– A numerical scheme is said to be convergent if the solution of the discretized equations tend to the exact solution of the (P)DE as the grid-spacing and time-step go to zero

– Error equation for linear(-ized) systems:

– Error bounds for linear systems:

Convergence <= Stability + Consistency (for linear systems)

= Lax Equivalence Theorem (for linear systems)

– For nonlinear equations, numerical experiments are often used

• e.g., iterate or approximate true solution with computation on successively finer grids, and compute resulting discretization errors and order of convergence

1ˆ ( )x x

1ˆ ( )x x( )x x( ) 1 1( ) ( )

( ) ( )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

1 1

1

ˆ ˆ( )

For a consistent scheme: ( ) for 0ˆHence ( )

x x x x

px

px x

O x x

O x

1 11 11 11 11 1ˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆ1 1ˆ ˆ1 11 1ˆ ˆ1 11 1ˆ ˆ1 11 1ˆ ˆ1 11 1ˆ ˆ1 11 1( )1 1x x x xx x x xx x x xx x x xx x x xx x x xx x x xx x x xx x x x 1 1 1 11 1 1 11 1 1 1 1 1 1 11 1 1 11 1 1 1

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 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( )x x x x( ) ( )x x x x( ) ( ) ( ) ( ) ( )x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x xx 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( ) 1 1 1 11 1 1 11 1 1 11 1( )1 1 1 1( )1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1( ) ( ) ( ) ( )1 1( )1 1 1 1( )1 1 1 1( )1 1 1 1( )1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1( )1 1 1 1( )1 1 1 1( )1 1 1 1( )1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1ˆHence ( )Hence ( )Hence ( )Hence ( )Hence ( )Hence ( )1Hence ( )1Hence ( )Hence ( )x xx xx xx xx xHence ( )x xHence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )x x x xx x x xx x x xx x x xx x x xHence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( ) Hence ( )x xHence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )Hence ( ) Hence ( )1Hence ( )1 1Hence ( )1Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( )1Hence ( )1 1Hence ( )1 1Hence ( )1 1Hence ( )1Hence ( )Hence ( ) Hence ( )Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( )Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( ) Hence ( )


Convergence


Finite Differences - Basics

• Finite Difference Approximation idea directly borrowed

from the definition of a derivative.

• Geometrical Interpretation

10

( ) ( )'( ) lim i ii x

x xxx

– Quality of approximation

improves as stencil points

get closer to xi

– Central difference would be

exact if was a second

order polynomial and points

were equally spaced ExactBackward

CentralForward

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

φ

∆xi∆xi+1

Image by MIT OpenCourseWare.

On the definition of a derivative and its approximations


FINITE DIFFERENCES:

Taylor Series, Higher Order Accuracy

1) First approach: Use Taylor series, keep more higher-order terms

than strictly needed and express these higher-order terms as

finite-differences themselves

• For example, how can we derive the forward finite-difference

estimate of the first derivative at xi with second order accuracy?

• If we retain the second-derivative, and estimate it with first-order

accuracy, the order of accuracy for the estimate of f’(xi) will be p=2

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

How to obtain differentiation formulas of arbitrary high accuracy?

23

1( ) ( ) '( ) ''( ) ( )2!i i i ixf x f x x f x f x O x

21( ) ( )'( ) ''( ) ( )2!

i ii i

f x f x xf x f x O xx


FINITE DIFFERENCES:

Taylor Series, Higher Order Accuracy Cont’d

Still using

• Estimate the second-derivative with forward finite-differences at first-

order accuracy:

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

23

1

23

2

( ) ( ) '( ) ''( ) ( )2!4( ) ( ) 2 '( ) ''( ) ( )

2!

i i i i

i i i i

xf x f x x f x f x O x

xf x f x x f x f x O x

21

2 21 2 1 2 12

( ) ( )'( ) ''( ) ( )2!

( ) ( ) ( ) 2 ( ) ( ) ( ) 4 ( ) 3 ( )'( ) ( ) ( )2! 2

i ii i

i i i i i i i ii

f x f x xf x f x O xx

f x f x x f x f x f x f x f x f xf x O x O xx x x

2 12

* ( 2) ( ) 2 ( ) ( )''( ) ( )* (1)

i i ii

f x f x f xf x O xx


Figure 23.1

Chapra and

Canale

Forward

Differences


Backward

Differences


Centered

Differences


Problem: Estimate 1st derivative of f = -0.1*x^4 - 0.15*x^3-0.5*x^2-0.25*x +1.2 at

x=0.5, with a grid cell size of h=0.25 and using successively higher order

schemes. How does the solution improve?

%Define the function

f=@(x) -0.1*x^4 - 0.15*x^3-0.5*x^2-0.25*x +1.2;

%Define Step size

h=0.25;

%Set point at which to evaluate the derivative

x = 0.5;

%% Using forward difference

%First order:

df=(f(x+h)-f(x)) / h;

fprintf('\n\n First order Forward difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

%Second order:

df=(-f(x+2*h)+4*f(x+h)-3*f(x)) / (2*h);

fprintf('Second order Forward difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

%% Backwards difference

%First order:

df=(-f(x-h)+f(x)) / (h);

fprintf('First order Backwards difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

%Second order:

df=(f(x-2*h)-4*f(x-h)+3*f(x)) / (2*h);

fprintf('Second order Backwards difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

%% Central difference

%Second order:

df=(f(x+h)-f(x-h)) / (2*h);

fprintf('Second order Central difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

%Fourth order:

df=(-f(x+2*h)+8*f(x+h)-8*f(x-h)+f(x-2*h)) /

(12*h);

fprintf('Fourth order Central difference: %g, with

error:%g%% \n',df,abs(100*(df+0.9125)/0.9125))

FINITE DIFFERENCES

Taylor Series, Higher Order Accuracy: EXAMPLE

L11_FD.m

First order Forward difference: -1.15469, with error:26.5411%

Second order Forward difference: -0.859375, with error:5.82192%

First order Backwards difference: -0.714063, with error:21.7466%

Second order Backwards difference: -0.878125, with error:3.76712%

Second order Central difference: -0.934375, with error:2.39726%

Fourth order Central difference: -0.9125, with error:2.43337e-14%

Why is the 4th order “exact”?

Output


• 1st Approach:

– Incorporate more higher-order terms of the Taylor series expansion

than strictly needed and express them as finite differences themselves

– e.g. for finite difference of mth derivative at order of accuracy p, express

the m+1th, m+2th, m+p-1th derivatives at an order of accuracy p-1, .., 2, 1.

– General approximation:

– Can be used for forward, backward, skewed or central differences

– Can be computer automated

– Independent of coordinate system and extends to multi-dimensional

finite differences (each coordinate is often treated separately)

• Remember: order p of approximation indicates how fast the error is

reduced when the grid is refined (not necessarily the magnitude of

the error)

FINITE DIFFERENCES:

Taylor Series, Higher Order Accuracy

Summary

m s

i j i xmi rj

u a ux


FINITE DIFFERENCES:

Interpolation Formulas for Higher Order Accuracy

2nd approach: Generalize Taylor series using interpolation formulas

• Fit the unknown function solution of the (P)DE to an interpolation curve

and differentiate the resulting curve. For example:

• Fit a parabola to “f data” at points , then

differentiate to obtain:

• This is a 2nd order approximation (parabola approx. is of order 3)

• For uniform spacing, reduces to centered difference seen before

• In general, approximation of first derivative has a truncation error of the order

of the polynomial (here 2)

• All types of polynomials or numerical differentiation methods can be

used to derive such interpolations formulas

• Polynomial fitting, Method of undetermined coefficients, Newton’s

interpolating polynomials, Lagrangian and Hermite Polynomials, etc

( ( ( ( 2 2 2 2

1 1 1 1

1 1

( ) ( ) ( )'( )

( )i i i i i i i

ii i i i

f x x f x x f x x xf x

x x x x

1 1 1, , ( )i i i i i ix x x x x x


FINITE DIFFERENCES Higher Order Accuracy: Taylor Tables or Method of Undetermined Coefficients

Taylor Tables: Convenient way of forming linear combinations of Taylor Series

on a term-by-term basis

Taylor series at:

j-1

j

j+1

Sum each column starting from left, force the sums to zero and so choose a, b, c, etc

What we are looking for, in1st column:


FINITE DIFFERENCES

Higher Order Accuracy: Taylor Tables Cont’d

Sum each column starting from left and force the sums to be zero by proper choice of a, b, c, etc:

1 1 1 01 0 1 0 1 2 11 0 1 2

ab a b cc

Truncation error is first column in the table that does not vanish, here fifth column of table:

4 2 44

2 4 4

124 24 12x

j j

a c u x uxx x x

= Familiar 3-point

central difference


FINITE DIFFERENCES

Higher Order Accuracy: Taylor Tables Cont’d

2 1 1 4 3 / 2a a b

(as before)

3 2 332 1

3 3

1 86 6 3x

j j

a a u x uxx x x

and

2.29 Numerical Fluid Mechanics PFJL Lecture 8 N-S, 24

Integral Conservation Law for a scalar

fixed fixed

Advective fluxes Other transports (diffusion, etc)Sum of sources and(Adv.& diff. fluxes convection)sinks terms (r

( . ) .CM CV CS CS CV

d ddV dV v n dA q n dA s dVdt dt

dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .

fixed fixed

Advective fluxes Other transports (diffusion, etc)fixed fixedCM CV CS CS CVfixed fixedCM CV CS CS CV CM CV CS CS CVfixed fixedCM CV CS CS CVfixed fixed

fixed fixedCM CV CS CS CVfixed fixed

dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV

dV dV v n dA q n dA s dVCM CV CS CS CV CM CV CS CS CV

dV dV v n dA q n dA s dVCM CV CS CS CV


( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV

( . ) . ( . ) .CM CV CS CS CV


( . ) .CM CV CS CS CV CM CV CS CS CV CM CV CS CS CV CM CV CS CS CV

dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV







( . ) . ( . ) .CM CV CS CS CV


( . ) . ( . ) .CM CV CS CS CV


( . ) . ( . ) .CM CV CS CS CV



CM CV CS CS CV CM CV CS CS CV

CM CV CS CS CV

dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVCM CV CS CS CV


CM CV CS CS CV



CM CV CS CS CV


dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) . ( . ) .dV dV v n dA q n dA s dV( . ) .

eactions, etc)

fixed fixed

Sum of sources and

CM CV CS CS CVfixed fixedCM CV CS CS CVfixed fixed

dV dV v n dA q n dA s dV CM CV CS CS CV CM CV CS CS CVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV

CM CV CS CS CVdV dV v n dA q n dA s dV

CM CV CS CS CV CM CV CS CS CVdV dV v n dA q n dA s dV

CM CV CS CS CVCM CV CS CS CV CM CV CS CS CVfixed fixedCM CV CS CS CVfixed fixed

fixed fixedCM CV CS CS CVfixed fixedCM CV CS CS CV CM CV CS CS CV CM CV CS CS CV CM CV CS CS CVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV





CM CV CS CS CVCM CV CS CS CV CM CV CS CS CVCM CV CS CS CV CM CV CS CS CV CM CV CS CS CV CM CV CS CS CVCM CV CS CS CV CM CV CS CS CVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dV dV dV v n dA q n dA s dVdV dV v n dA q n dA s dV dV dV v n dA q n dA s dV



CM CV CS CS CVCM CV CS CS CVdV dV v n dA q n dA s dV




CM CV CS CS CVCM CV CS CS CVdV dV v n dA q n dA s dV


CM CV CS CS CV

. ( ) . ( )v k st

. ( ) . ( )v k s. ( ) . ( ) . ( ) . ( )v k s. ( ) . ( ). ( ) . ( )v k s. ( ) . ( ) . ( ) . ( )v k s. ( ) . ( ) . ( ) . ( )v k s. ( ) . ( ) . ( ) . ( )v k s. ( ) . ( )

CV,fixed sΦ

qq

ρ,Φ vv

.( ) .v q st

v q s v q s v q s.( ) ..( ) ..( ) . .( ) . .( ) .v q s v q s.( ) .v q s.( ) . .( ) .v q s.( ) ..( ) ..( ) . .( ) ..( ) ..( ) .v q s.( ) ..( ) .v q s.( ) . .( ) .v q s.( ) ..( ) .v q s.( ) . v q s v q s

Applying the Gauss Theorem, for any arbitrary CV gives:

For a common diffusive flux model (Fick’s law, Fourier’s law):

q k q k q k q k

Conservative form of the PDE

Strong-Conservative form

of the Navier-Stokes Equations ( v)

2.29 Numerical Fluid Mechanics PFJL Lecture 8-NS, 25

CV,fixed sΦ

qq

ρ,Φ vv.( ) .v v v p g

t

.( ) .v .( ) .v v p gv v p g v v p g .( ) . .( ) . .( ) . .( ) . .( ) . .( ) .v v p g v v p g v v p g v v p g.( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) ..( ) .v v p g v v p g v v p g v v p g v v p g v v p g v v p g.( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) ..( ) ..( ) . .( ) ..( ) . .( ) . .( ) . .( ) ..( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) . .( ) .v v p g.( ) .

Applying the Gauss Theorem gives:

Equations are said to be in “strong conservative form” if all terms have the form of the divergence

of a vector or a tensor. For the i th Cartesian component, in the general Newtonian fluid case:

(

( . ) .

.

CV CS CS CS CV

F

CV

d vdV v v n dA p ndA ndA gdVdt

p g dV

F

vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .CV CS CS CS CV

F

vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdVCV CS CS CS CV

vdV v v n dA p ndA ndA gdVCV CS CS CS CV

CV CS CS CS CVvdV v v n dA p ndA ndA gdV

CV CS CS CS CV

CV CS CS CS CV CV CS CS CS CVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .


CV CS CS CS CV


CV CS CS CS CV CV CS CS CS CVvdV v v n dA p ndA ndA gdV

CV CS CS CS CV


CV CS CS CS CV( . ) .

CV CS CS CS CV( . ) .vdV v v n dA p ndA ndA gdV( . ) .

CV CS CS CS CV( . ) . ( . ) .


CV CS CS CS CV( . ) . ( . ) .


CV CS CS CS CV( . ) . ( . ) .


CV CS CS CS CV( . ) .vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .

CV CS CS CS CV

F

vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdVvdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV vdV v v n dA p ndA ndA gdV( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) . ( . ) .vdV v v n dA p ndA ndA gdV( . ) .

F

p g dVp g dVp g dV p g dVp g dV p g dV p g dV p g dVp g dVp g dVp g dV p g dV

p g dV p g dVp g dV p g dV p g dV p g dV

With Newtonian fluid + incompressible + constant μ:

For any arbitrary CV gives:

2.( )

. 0

v v v p v gtv

.( )v 2.( )v v p v g v v p v g v v p v g 2 2.( ) .( ) 2 2 2 2v v p v g v v p v g v v p v g v v p v g.( ) .( ) .( ) .( ).( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( ).( )v v p v g v v p v g v v p v g v v p v g.( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( ).( ) .( ) .( ) .( ) .( ) .( ) .( )v v p v g v v p v g v v p v g v v p v g.( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( ) .( )v v p v g.( )

. 0t

. 0 . 0. 0v. 0 . 0v. 0

Momentum:

Mass:

2.( ) .3

j ji ii i j i i i i

j i j

u uv uv v p e e e g x et x x x

u u u u v u v u v u v u 2 2 2 2u u u u u u u u2u u2 2u u2 2u u2 2u u2 u u u u u u u u u u u u u u u u v u v u v u v u v u v u v u v u v u v u v u v u

j j

j j2j j2

2j j2j ju uj j

j ju uj jv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

j j

j j

j j

j ju u u u

u u u uj ju uj j

j ju uj j

j ju uj j

j ju uj j

v u v u

v u v u v u v u v u v u

v u v u v u v u j j j j

j j j j2j j2 2j j2

2j j2 2j j2j ju uj j j ju uj j

j ju uj j j ju uj j2j j2u u2j j2 2j j2u u2j j2

2j j2u u2j j2 2j j2u u2j j2u u u u u u u u

u u u u u u u u2u u2 2u u2 2u u2 2u u2

2u u2 2u u2 2u u2 2u u2j ju uj j

j ju uj j j ju uj j

j ju uj j

j ju uj j

j ju uj j j ju uj j

j ju uj j2j j2u u2j j2 2j j2u u2j j2 2j j2u u2j j2 2j j2u u2j j2

2j j2u u2j j2 2j j2u u2j j2 2j j2u u2j j2 2j j2u u2j j2

u u u u u u u u

u u u u u u u uj ju uj j j ju uj j j ju uj j j ju uj j

j ju uj j j ju uj j j ju uj j j ju uj j u u u u u u u u u u u u u u u u

u u u u u u u u u u u u u u u uj ju uj j

j ju uj j j ju uj j

j ju uj j j ju uj j

j ju uj j j ju uj j

j ju uj j

j ju uj j

j ju uj j j ju uj j

j ju uj j j ju uj j

j ju uj j j ju uj j

j ju uj jv u v u

v u v ui iv ui i i iv ui i i iv ui i i iv ui iv u v u v u v u

v u v u v u v uv u v u v u v u

v u v u v u v uv u v u v u v u v u v u v u v u

v u v u v u v u v u v u v u v u

j j

j j

j j

j j2j j2

2j j2

2j j2

2j j2 j j j j

j j j j

j j j j

j j j jv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ej jv v p e e e g x ej j j jv v p e e e g x ej j

j jv v p e e e g x ej j j jv v p e e e g x ej j


j jv v p e e e g x ej j j jv v p e e e g x ej ji i i i i i i ii i i i i i i i i i i i i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei ii i i i i i i i i i i i i i i ii iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i j j

j j

j j

j ji i i i

i i i ii iv ui i i iv ui i

i iv ui i i iv ui i

j j j j

j j j j

j j j j

j j j ji i

i i i i

i i

i i

i i i i

i iv u v u

v u v u

v u v u

v u v ui iv ui i

i iv ui i i iv ui i

i iv ui i

i iv ui i

i iv ui i i iv ui i

i iv ui i

j j j j j j j j

j j j j j j j j

j j j j j j j j

j j j j j j j ju u u u u u u u

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i i i i i i i i i i i i i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ej jv v p e e e g x ej j j jv v p e e e g x ej j







j jv v p e e e g x ej j j jv v p e e e g x ej ji iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i

i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei ii i i i i i i i i i i i i i i i

i i i i i i i i i i i i i i i ii iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i

i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ej jv v p e e e g x ej j j jv v p e e e g x ej j















j jv v p e e e g x ej j j jv v p e e e g x ej ji iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i

i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i



i i j i i i i i i j i i i i i i j i i i i i i j i i i iv v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ev v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ev v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e


v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ev v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e



v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ev v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e



v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i

v u v ui iv ui i i iv ui i.( ) .i i.( ) .v u.( ) .i i.( ) . .( ) .i i.( ) .v u.( ) .i i.( ) . v u v u v u v ui i i i.( ) .i i.( ) . .( ) .i i.( ) .i i i i i i i i.( ) .i i.( ) . .( ) .i i.( ) . .( ) .i i.( ) . .( ) .i i.( ) .v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e.( ) .v v p e e e g x e.( ) . .( ) .v v p e e e g x e.( ) . .( ) .v v p e e e g x e.( ) . .( ) .v v p e e e g x e.( ) .i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i.( ) .i i.( ) .v v p e e e g x e.( ) .i i.( ) . .( ) .i i.( ) .v v p e e e g x e.( ) .i i.( ) . .( ) .i i.( ) .v v p e e e g x e.( ) .i i.( ) . .( ) .i i.( ) .v v p e e e g x e.( ) .i i.( ) .i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i.( ) .i i j i i i i.( ) .v v p e e e g x e.( ) .i i j i i i i.( ) . .( ) .i i j i i i i.( ) .v v p e e e g x e.( ) .i i j i i i i.( ) . .( ) .i i j i i i i.( ) .v v p e e e g x e.( ) .i i j i i i i.( ) . .( ) .i i j i i i i.( ) .v v p e e e g x e.( ) .i i j i i i i.( ) .i i i i i i i i.( ) .i i.( ) . .( ) .i i.( ) . .( ) .i i.( ) . .( ) .i i.( ) .i iv ui i i iv ui i i iv ui i i iv ui i.( ) .i i.( ) .v u.( ) .i i.( ) . .( ) .i i.( ) .v u.( ) .i i.( ) . .( ) .i i.( ) .v u.( ) .i i.( ) . .( ) .i i.( ) .v u.( ) .i i.( ) .v u v u

v u v ui iv ui i i iv ui i i iv ui i i iv ui iv u v u v u v u

v u v u v u v ui i i i i i i ii i i i i i i i i i i i i i i iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei i i iv v p e e e g x ei ii i i i i i i i i i i i i i i ii iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui i i iv ui iv v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e

v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x e v v p e e e g x ei i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i i i i j i i i iv v p e e e g x ei i j i i i iWith Newtonian fluid only:

Cons. of Momentum:

Cauchy Mom. Eqn.

MIT OpenCourseWarehttp://ocw.mit.edu

2.29 Numerical Fluid MechanicsSpring 2015

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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end of (linear) algebraic systems - mit …...2.29 numerical fluid mechanics pfjl lecture 10, 1 2.29...

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