End of (Linear) Algebraic Systems - MIT …...2.29 Numerical Fluid Mechanics PFJL Lecture 10, 1 2.29...
Transcript of End of (Linear) Algebraic Systems - MIT …...2.29 Numerical Fluid Mechanics PFJL Lecture 10, 1 2.29...
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
PFJL Lecture 10, 2Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 3Numerical 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”
PFJL Lecture 10, 4Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 5Numerical Fluid Mechanics2.29
Partial Differential Equations
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
PFJL Lecture 10, 6Numerical Fluid Mechanics2.29
t
xu(x,0), ut(x,0)
u(0,t) u(L,t)
Partial Differential Equations
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:
PFJL Lecture 10, 7Numerical Fluid Mechanics2.29
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:
Partial Differential Equations
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
(
Error Types and Discretization Properties:
T
2.29 Numerical Fluid Mechanics PFJL Lecture 10, 9
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
Error Types and Discretization Properties:
S
2.29 Numerical Fluid Mechanics PFJL Lecture 10, 10
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.
PFJL Lecture 10, 11Numerical Fluid Mechanics2.29
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 ( )
Error Types and Discretization Properties:
Convergence
PFJL Lecture 10, 12Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 13Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 14Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 15Numerical Fluid Mechanics2.29
Figure 23.1
Chapra and
Canale
Forward
Differences
PFJL Lecture 10, 16Numerical Fluid Mechanics2.29
Backward
Differences
PFJL Lecture 10, 17Numerical Fluid Mechanics2.29
Centered
Differences
PFJL Lecture 10, 18Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 19Numerical Fluid Mechanics2.29
• 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
PFJL Lecture 10, 20Numerical Fluid Mechanics2.29
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
2.29 Numerical Fluid Mechanics PFJL Lecture 10, 21
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:
PFJL Lecture 10, 22Numerical Fluid Mechanics2.29
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
PFJL Lecture 10, 23Numerical Fluid Mechanics2.29
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
( . ) .CM CV CS CS CV
( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV
( . ) . ( . ) .CM 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
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
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 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
( . ) .CM CV CS CS CV
( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV
( . ) . ( . ) .CM CV CS CS CV
( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV
( . ) . ( . ) .CM CV CS CS CV
( . ) .dV dV v n dA q n dA s dV( . ) .CM CV CS CS CV
( . ) . ( . ) .CM 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
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
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 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 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 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 CV 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 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 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 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 CV 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 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 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
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
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 CVvdV v v n dA p ndA ndA gdV
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 CVvdV v v n dA p ndA ndA gdV
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( . ) .vdV v v n dA p ndA ndA gdV( . ) .
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( . ) .vdV v v n dA p ndA ndA gdV( . ) .
CV CS CS CS CV( . ) . ( . ) .
CV CS CS CS CV( . ) .vdV v v n dA p ndA ndA gdV( . ) .
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
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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 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 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 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 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
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