MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.29 NUMERICAL FLUID MECHANICS — SPRING 2015
EQUATION SHEET – Quiz 1
Number Representation
- Floating Number Representation: x = m be , b-1 ≤ m < b0
Truncation Errors and Error Analysis y  f ( x1, x2 , x3 ,..., xn )
- Taylor Series:
f ( xi 1 )  f ( xi )  x f '( xi ) 
Rn 
x 2
x 3
x n n
f ''( xi ) 
f '''( xi )  ... 
f ( xi )  Rn
2!
3!
n!
x n 1 ( n 1)
f
( )
n  1!
n
- The Differential Error (general error propagation) Formula:  y  
i 1
- The Standard Error (statistical formula): E (  s y )
- Condition Number of f(x): K p 
f ( x1 ,..., xn )
i
xi
2
 f  2


 i
i 1  xi 
n
x f '( x )
f (x)
Roots of nonlinear equations ( xn1  xn  h( xn ) f ( xn ) )
- Bisection: successive division of bracket in half, next bracket based on sign of
n 1
f (x1n1 ) f (xmid-point
)
f ( xU ) ( xL  xU )
f ( xL )  f ( xU )
- Fixed Point Iteration (General Method or Picard Iteration):
- False-Position (Regula Falsi): xr  xU 
xn1  g ( xn ) or xn1  xn  h( xn ) f ( xn )
1
f ( xn )
f '( xn )
( xn  xn 1 )
- Secant Method: xn 1  xn 
f ( xn )
f ( xn )  f ( xn 1 )
- Newton Raphson: xn 1  xn 
- Order of convergence p: Defining en  xn  x e , the order of convergence p exists if there
exist a constant C≠0 such that: lim
n 
en 1
en
p
C
-1-
Conservation Law for a scalar , in integral and differential forms:
d
 d
-    dV  
 dV  CS  (v.n )dA   CS q .n dA 
CM
 dt
 dt CVfixed
Advective fluxes
(Adv.& diff. ="convection" fluxes)
-
Other transports
(diffusion, etc)

CVfixed
s dV
Sum of sources and
sinks terms (reactions, etc)

 .( v )  . q  s
t
Linear Algebraic Systems:
aik( k )
( k 1)
(k )
(k )
( k 1)
 bi( k )  mik bk( k ) ,
- Gauss Elimination: reduction, mik  ( k ) , aij  aij  mik akj , bi
akk

followed by a back-substitution. xk   bk 

- LU decomposition: A=LU, aij 
min( i , j )

k 1
n
a
j  k 1
(k )
kj

x j  akk( k )

mik ak( kj )
- Choleski Factorization: A=R*R, where R is upper triangular and R* its conjugate transpose.
- Condition number of a linear algebraic system: K ( A)  A 1 A
- A banded matrix of p super-diagonals and q sub-diagonals has a bandwidth w = p + q + 1
n
n
n(n  1)
n(n  1)(2n  1)
- i 
and  i 2 
2
6
i 1
i 1
- Eigendecomposition: Ax  x and det(A  I)  0
- Norms:
m
A 1  max  aij
1 j  n
A
A
n

F
 max  aij
1 i  m

“Maximum Column Sum”
i 1
“Maximum Row Sum”
j 1
2
 m n

  aij 
 i 1 j 1

A 2  max A*A
“Frobenius norm”  or “Euclidean norm” 
“ L  2 norm”  or “spectral norm” 
Iterative Methods for solving linear algebraic systems: xk 1  B x k  c
- Necessary and sufficient condition for convergence:
 (B)  max i  1, where i  eigenvalue(Bnn )
k  0,1,2,...
i 1... n
- Jacobi’s method: xk 1  -D-1 (L  U) x k  D-1b
- Gauss-Seidel method: xk 1  (D  L)-1 U xk  (D  L)-1 b
- SOR Method: xk 1  (D  L)1[U  (1  )D]x k  (D  L)1 b
ri T ri
ri , ri  b  Axi
ri T Ari
- Conjugate Gradient: xi 1  xi  i vi (αi such that each vi are generated by orthogonalization
of residuum vectors and such that search directions are A-conjugate).
- Steepest Descent Gradient Method: xi 1  xi 
-2-
Finite Differences – PDE types (2nd order, 2D): Axx  Bxy  Cyy  F ( x, y,  , x , y )
B2  AC > 0: hyperbolic;
B2  AC = 0: parabolic;
B2  AC < 0: elliptic
Finite Differences – Error Types and Discretization Properties (
- Consistency:
( )  ˆx ( )  0 when x  0
- Truncation error:  x 
- Error equation:  x 
- Stability:
( )  0 ,
( )  ˆx ( )  O(x p ) for x  0
( )  ˆx (ˆ   )   ˆx ( ) (for linear systems)
ˆ 1  Const.
x
- Convergence:  
ˆ 1
x
(for linear systems)
 x   O(x p )
-3-
ˆ (ˆ)  0 )
x
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2.29 Numerical Fluid Mechanics
Spring 2015
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