r Parabolic,DEs

(4-1)

(4-2)

ui 1–n+ )

∆ t

x)2---------

12---≤

Page 4-1

Chapter 4 Numerical Techniques foElliptic, and Hyperbolic P

4-1 Parabolic PDEs in 1D

(1) Explicit Methods

(a) FTCS

(b) The DuFort-Frankel Scheme

CTCS but in the diffusion term is replaced by

t∂∂u α

x2

2

∂∂ u

=

uin 1+ ui

n α∆ t

∆x( )2-------------- ui 1+

n 2uin–(+=

O ∆t( ) ∆x( )2,[ ]

Stability Requiresα∆(

-----i-1 i i+1

n

n+1

uin

uin

uin 1+ ui

n 1–+

2---------------------------------= ?????

(4-3)

(4-4)

called time linear extrapolation?

t)] ∆t( ) O ∆t( )2[ ]+

)2]

1-----

1+ ui 1–n+ )

Page 4-2

Digression:

Accordingly,

or

uin

uin 1+ ui

n 1–+

2-----------------------------------= ui

n 1+ uin–⇒ ui

n uin 1––= ???

uin 1+ ui

nt∂∂u

i

nO ∆t( )2[ ]+ + ui

nui

n uin 1––( )

∆t------------------------------- O ∆([++= =

⇒ uin 1+ ui

n– uin ui

n 1–– O ∆t([+=

uin 1+ ui

n 1––

2 ∆t( )---------------------------------- α

ui 1+n 2

uin 1+ ui

n 1–+

2-----------------------------------– ui –

n+

∆x( )2------------------------------------------------------------------------=

12α ∆ t( )∆x( )2

------------------+ uin 1+ 1

2α ∆ t( )∆x( )2

------------------– uin 1– 2α ∆ t( )

∆x( )2------------------ ui

n(+=

(4-5)

e rate, such that

ol. to start with and 2

(4-3)/(4-4) gives

x)2 O ∆x( )4[ ]+t

n

tx

---- 0⇒

αt2

2

∂∂ u

A Hyp. Eq.=

Page 4-3

Discussions:

(a)

(b)Unconditionally stable(c)Consistency (??)

Note: If and were to approach zero at the sam

(d) Not a self-starting method, need 2 time level stime level storage space

O ∆t( )2 ∆x( )2 ∆t∆x------ 2

, ,

uin 1+ in time about ui

n uin 1– in time about ui

n

ui 1+n in space about ui

n ui 1–n in space about ui

n

into⇒

t∂∂u α

t2

2

∂∂ u

∆t

∆x------ 2 1

6---

t3

3

∂∂ u

∆t( )2 O ∆t( )3[ ]+ ++ αt2

2

∂∂ u α

12------

x4

4

∂∂ u ∆(+=

4 5–( )[ ]∆x 0→∆t 0→

lim PDE⇒ Only if∆∆--

∆x 0→∆t 0→

lim

∆t ∆x

∆t∆x------

∆x 0→∆t 0→

lim K⇒ then 4 5–( ) becomest∂∂u αK

t2

2

∂∂ u

+

icit Scheme)

(4-6)

(4-7)ui

n 1+ uin–

β 1≤ ≤

∆t( )2[ ]

Page 4-4

(2) Implicit Methods

(a) BTCS (Fully/Euler Implicit Scheme)

(b) Crank-Nicolson Scheme (Trapezoidal Impl

(c) One-Parameter Formulation

then if

NOTE: If

t∂∂u

F r u …, ,( )= e.g., Fx

2

2

∂

∂ u=

∆uin

∆t---------- β

t∂∂u

i

n 1+1 β–( )

t∂∂u

i

n+= where

∆uin =

0

β 1 Euler Implicit⇒= O ∆t( )[ ]

β 12--- Crank-Nicolson Implicit⇒= O

β 0 Euler Explicit⇒= O ∆t( )[ ]12--- β 1 Unconditionally Stable⇒≤ ≤

0 β 12--- Conditionally Stable⇒≤ ≤

(4-8)

(4-9)

er/nonlinear in u):

∂∂

linearization, e.g.,

nO ∆t( )2[ ]+

E Au∂∂E

and Bu∂∂F

= =

n Bn∆un O ∆t( )2[ ]+ +

Page 4-5

Accordingly

e.g.,

d) Delta (∆) Formulation

Define

For the following problem (fluxes E and F may be lin

uin 1+ ui

n–

∆t-------------------------- βFi

n 1+ 1 β–( )Fin+=

uin 1+ ui

n–

∆t-------------------------- α β

x2

2

∂

∂ u

i

n 1+

1 β–( )x

2

2

∂

∂ u

i

n

+=

∆un un 1+ un–=

tu

x∂∂E

x2

2

∂∂ F

+= where E and F = fn(t,x,u, ... ) then

En 1+ En ∆tt∂∂E n

O ∆t( )2[ ]+ + En ∆tu∂∂E

t∂∂u += =

n 1+ En ∆t An ∆un

∆t---------- O ∆t( )[ ]+ O ∆t( )2[ ]+ += where

En 1+ En An∆un O ∆t( )2[ ]+ += similarly Fn 1+ F=

(4-10)

(4-11)

x∂∂

Enx∂∂

Fn+

x∂∂

Fn+

Fn

)x∂∂

x∂∂

un

u …⇒

γ2– O ∆t( )2[ ]

Page 4-6

e.gs.,

e) Two-Parameter Formulation

un 1+ un–∆t

-------------------------- βx∂∂

En An∆unx∂∂

Fn Bn∆un+( )+ + 1 β–( )+=

βx∂∂

An∆unx∂∂

Bn∆un( )+x∂∂

Enx∂∂

Fn+ +=

⇒ ∆un β ∆t( )x∂∂

An∆unx∂∂

Bn∆un( )+– ∆t( )x∂∂

En=

1 β ∆t( )x∂∂

Anx∂∂

Bn+ – ∆un ∆t( )

x∂∂

Enx∂∂

+=

t∂∂u α

x2

2

∂∂ u

= 1 βα ∆t( )x∂∂

x∂∂

– ∆un α ∆ t(=⇒

t∂∂u

ux∂∂u

+t∂∂u

x∂∂ u2

2----- += 0= E⇒ u2

2-----= A⇒ =

γ1∆t

1 γ2+--------------

t∂∂u

–

n 1+∆un+

1 γ– 1( )∆t

1 γ2+-------------------------

t∂∂u

n γ21 γ2+--------------∆un 1– γ1

12---–

+ +=

O ∆t( )2[ ] if γ1 γ212---+=

4

(4-12)

Accuracy in Time

O ∆t( )[ ]

O ∆t( )2[ ]

O ∆t( )2[ ]

O ∆t( )[ ]

O ∆t( )2[ ]

Page 4-7

-2 Multi-Dimensional Parabolic PDEs

e.g.,

Scheme

0 0 Euler Explicit Scheme

0 - 0.5 Explicit Leapfrog Scheme

0.5 0 Implicit Trapezoidal Scheme (C-N)

1 0 Euler Implicit Scheme

1 0.5 Three-Point Backward Implicit Scheme

γ1 γ2

t∂∂u α

x2

2

∂∂ u

y2

2

∂∂ u

+

=

(4-13)i j, 1–n

---------------

i

j

n

x

y

α∆ t

∆y( )2--------------

12--- or d

14---≤

Page 4-8

(1) Explicit: FTCS

(a)

(b) Stability:

Note:

Inefficient and unattractive

Implicit Methods

ui j,n 1+ ui j,

n–

∆t----------------------------- α

ui 1 j,+n 2ui j,

n– ui 1 j,–n+

∆x( )2-----------------------------------------------------------

ui j, 1+n 2ui j,

n– u+

∆y( )2--------------------------------------------+=

∆y∆x

i-1,j i,j i+1,j

i,j-1

i,j+1

x

y

t

O ∆t( ) ∆x( )2 ∆y( )2, ,[ ]

dx dy+( ) 12---≤ dx

α∆ t

∆x( )2--------------= dy =

if ∆x ∆y The diffusion number dx⇒ dy d 2d ≤⇒= = =

⇒

⇒

(4-14)

(4-15)

(4-16)

ed Matrix

j1+ ui j, 1–

n 1++

)2------------------------------

i j 1+,n 1+ ui j,

n–=

1+1 fi j,=

.......................

........................

............................ .

.

.

imax - 1 spaces

Page 4-9

(1) Implicit Methods:

(a) Fully Implicit (Euler Implicit)

or

[(imax-1)2 x (jmax-1)2] Pentadiagonal band

Inefficient and impreactical,

ADI (Alternating-Direction-Implicit),

AF (Approximate-Factorization),

Fractional Step,

etc.

ui j,n 1+ ui j,

n–

∆t----------------------------- α

ui 1 j,+n 1+ 2ui j,

n 1+– ui 1 j,–n 1++

∆x( )2-----------------------------------------------------------------

ui j, 1+n 1+ 2ui,

n–

∆y(-----------------------------------+=

⇒

dxui 1 j,+n 1+ dxui 1– j,

n 1+ 2dx 2dy 1+ +( )ui j,n 1+ dyui j 1–,

n 1+ dyu+ +–+

ai j, ui 1 j,+n 1+ bi j, ui 1– j,

n 1+ ci j, ui j,n 1+ di j, ui j 1–,

n 1+ ei j, ui j,n ++ + + +

⇒

⇒

⇒

⇒

⇒

⇒

..........

.

.

.

(4-17)

i j, ui j, 1–+ )n

y)2-----------------------------------

i j, ui j, 1–+ )n 1+

∆y)2-----------------------------------------

n+1

n+1/2

n

∆t/2

∆t/2

Page 4-10

(b) ADI (Alternating-Direction-Implicit)

Discussions:

(a)(b) Unconditionally Stable(c) Two sweeps, one along x, another

along y, each with tridiagonal matrix(d) BC for n+1/2 ? if time dependent.

ui j,n

12---+

ui j,n–

∆t2-----

----------------------------- αui 1 j,+ 2ui j,– ui 1 j,–+( )

n12---+

∆x( )2---------------------------------------------------------------------------

ui j, 1+ 2u–(

∆(-------------------------------+=

Solve for ui j,n

12---+

and then

ui j,n 1+ ui j,

n12---+

–

∆t2-----

----------------------------------- αui 1 j,+ 2ui j,– ui 1 j,–+( )

n12---+

∆x( )2---------------------------------------------------------------------------

ui j, 1+ 2u–(

(---------------------------------+=

to get ui j,n 1+( )

O ∆t( )2 ∆x( )2 ∆y( )2, ,[ ]

(4-18)

(4-19)

AF⇒

t∂∂u

i j,

n

α∆ t

∆x)2------------

α∆ t

∆y)2------------

j dyδy2ui j,

n+ )

Page 4-11

(c) AF (Approximate-Factorization)

Recall

Define

then for , we have

ADI AF⊂ and LU Decomposition

∆ui j,n

∆t------------- β

t∂∂u

i j,

n 1+1 β–( )

t∂∂u

i j,

n+=

⇒ ui j,n 1+ β ∆t( )

t∂∂u

i j,

n 1+– ui j,

n 1 β–( ) ∆t( )+=

δx2ui j, ui 1+ j, 2ui j,– ui 1– j,+= dx (

--=

δy2ui j, ui j 1+, 2ui j,– ui j 1–,+= dy (

--=

t∂∂u α

x2

2

∂∂ u

y2

2

∂∂ u

+

=

ui j,n 1+ β dxδx

2ui j,n 1+ dyδy

2ui j,n 1++( )– ui j,

n 1 β–( ) dxδx2ui,

n(+=

(4-20)

(4-21)

(4-22)

quence

(4-23)

(4-24)

(4-25)

ui j,*

i j,n 1+

Page 4-12

or

where

By AF, (4-20) becomes

(4-22) is solved according to the following se

Discussions:(a) (4-24) into (4-23), gives

1 β dxδx2 dyδy

2+( )–[ ] ui j,n 1+ RHS( )i j,

n=

RHS( )i j,n 1 1 β–( ) dxδx

2 dyδy2+( )+[ ] ui j,

n=

1 βdxδx2–( ) 1 βdyδy

2–( )ui j,n 1+ RHS( )i j,

n=

1 βdxδx2–( )ui j,

* RHS( )i j,n= Solve for

1 βdyδy2–( )ui j,

n 1+ ui j,*= Solve for u

1 βdxδx2–( ) 1 βdyδy

2–( )ui j,n 1+ RHS( )i j,

n=

(4-26)

(4-27)

(4-28)

(4-29)

(4-29) is

)i j,n

)dyδy2] ui j,

n

r! OK⇒

Page 4-13

or

(b)

(c) (4-22) can also be split as

Accordingly

NOTE: If ===> ADI, then the extra term in

(d) Delta Formulation!

1 βdxδx2 βdyδy

2 β2dxdyδx2δy

2+––[ ] ui j,n 1+ RHS(=

Extra Term which is at higher order => OK!

O ∆t( )2 ∆x( )2 ∆y( )2, ,[ ]

1 βdxδx2–( )ui j,

* 1 1 β–( )dyδy2+[ ] ui j,

n=

1 βdyδy2–( )ui j,

n 1+ 1 1 β–( )dxδx2+[ ] ui j,

*=

1 βdxδx2–( ) 1 βdyδy

2–( )ui j,n 1+ 1 1 β–( )dxδx

2+[ ] 1 1 β–(+[=

β 12---=

β2dxdyδx2δy

2 ui j,n 1+ ui j,

n–( ) Again, Higher Orde

1

(4-30)

type in x or y

Type in y or x

n

y2

2

∂∂ u

n

12---+

+

Page 4-14

(d) Fractional Step Methods

(a)

(b) Unconditionally Stable

1. Yanenko, N. N., 1971, The Method of Fractional Steps, translated by M. Holt in English, Springer-Verlag, N. Y.

n -> n+1/2 by CN

n+1/2 -> n+1 by CN

n

n+1/2

n+1

∆t2-----

∆t2-----

∆t

n n12---+→ ⇒ u

n12---+

un–

∆t2-----

--------------------------12---α

x2

2

∂∂ u

n

12---+

x2

2

∂∂ u

+=

n12---+ n 1+→ ⇒ un 1+ u

n12---+

–

∆t2-----

----------------------------------12---α

y2

2

∂∂ u

n 1+

=

O ∆t( )2 ∆x( )2 ∆y( )2, ,[ ]

(4-31)

(4-32)

13---∆t

Page 4-15

(e) Extension to Three Dimensional Problems

(I) ADI

(a)(b) Conditionally Stable

t∂∂u α

x2

2

∂∂ u

y2

2

∂∂ u

z2

2

∂∂ u

+ +

=

n n13---+ n

23---+ n 1+→ → → ⇒

un

13---+

un–∆t 3⁄( )

-------------------------- αx2

2

∂∂ u

n

13---+

y2

2

∂∂ u

n

z2

2

∂∂ u

n

+ +=

un

23---+

un

13---+

–∆t 3⁄( )

----------------------------------- αx2

2

∂∂ u

n

13---+

y2

2

∂∂ u

n

23---+

z2

2

∂∂ u

n

13---+

+ +=

un 1+ un

23---+

–∆t 3⁄( )

---------------------------------- αx2

2

∂∂ u

n

23---+

y2

2

∂∂ u

n

23---+

z2

2

∂∂ u

n 1+

+ +=

O ∆t( )2 ∆x( )2 ∆y( )2 ∆z( )2, , ,[ ]

dx dy dz+ +( ) 32---≤

(4-33)

z2

2u

n

n

z2

2

∂∂ u

n

+

z2

2

∂∂ u

n 1+

z2

2

∂∂ u

n

+

S)i j k, ,n

RHS)i j k, ,n

Page 4-16

(II) ADI + CN

(a)

(b) Unconditionally Stable

(III) AF

u* un–∆t

----------------- α 12---

x2

2

∂∂ u

*

x2

2

∂∂ u

n

+y2

2

∂∂ u

n

∂∂

+ +

=

u** un–∆t

-------------------- α 12---

x2

2

∂∂ u

*

x2

2

∂∂ u

n

+12---

y2

2

∂∂ u

**

y2

2

∂∂ u

++

=

un 1+ un–∆t

-------------------------- α 12---

x2

2

∂∂ u

*

x2

2

∂∂ u

n

+12---

y2

2

∂∂ u

**

y2

2

∂∂ u

n

+12---+ +

=

O ∆t( )2 ∆x( )2 ∆y( )2 ∆z( )2, , ,[ ]

1 β dxδx2 dyδy

2 dzδz2+ +( )–[ ] ui j k, ,

n 1+ RH(=

1 βdxδx2–( ) 1 βdyδy

2–( ) 1 βdzδz2–( )ui j k, ,

n 1+ (=

(4-34)

(4-35)ui j 1–,----------------- 0=

Page 4-17

4-2 Elliptic PDEse.g.

(1) FDE

Always Central Space

(a) 2nd-Order

•

x2

2

∂∂ u

y2

2

∂∂ u

+ 0= Laplaces Equation

x2

2

∂∂ u

y2

2

∂∂ u

+ F x y,( )= Poisson's Equation

ui 1 j,+ 2ui j,– ui 1 j,–+

∆x( )2----------------------------------------------------------

ui j 1+, 2ui j,– +

∆y( )2-----------------------------------------+∆y

∆x

i-1,j i,j i+1,j

i,j-1

i,j+1

xy

O ∆x( )2 ∆y( )2,[ ]

(4-36)

j--

i j, 2+--------------- 0=

Page 4-18

(a) 4th-Order (or higher)

•

• BC? ==> use smaller ∆x or ∆y grid.

• Rearrange (4-35)

ui 2 j,– 16u+–i 1 j,–

30ui j,– 16ui 1 j,+ ui 2,+–+

12 ∆x( )2--------------------------------------------------------------------------------------------------------------------------

ui j, 2– 16u+–i j, 1–

30ui j,– 16ui j, 1+ u–+

12 ∆y( )2-------------------------------------------------------------------------------------------------------------+

i-2,j i-1,j i,j i+1,j i+2,j

i,j+2

i,j+1

i,j-1

i,j-2

∆y

∆x

O ∆x( )4 ∆y( )4,[ ]

(4-37)

or small and dense,

for large, sparse

β ∆x∆y------=

.......

.........

..

.........

.

.

.

Page 4-19

• ==> Use ADI

(2) Solution Algorithms

(a) Iteration

(b) Direct (ROE??)

(c) Overview of Eq. Solving

(I) 1st Step:

Examine coefficient matrix A(nxn)

n is large or small

dense (few zero elements) or

sparse (many zero elements)

banded, diagonal, tridiagonal, etc.

(II) 2nd step

Decide which method to be used

direction methods: within finite steps. Good fbanded matrix

Iterative methods: approximation. Goodmatrix

β2ui j 1–, ui 1– j, 2 1 β2+( )ui j,– ui 1 j,+ β2u+ i j 1+,++ 0=

...

.......

.

.

..

root and f(x) = 0

x( ) ax3 bx2 cx d–+ +=

x) ax h bx( )tan–=

x( ) x2 7–=

*)x– )

----------

f x*( )f ' x*( )---------------–

g Iteration Function≡

x

x1

gy=x

Page 4-20

(d) Newton-Raphson Method: A Review

Find x such that f(x) = 0, e.g.,

Or TS Expansion about x*, where x* CLOSE to the

ax3 bx2 cx+ + d f⇒=

ax h bx( )tan f(⇒=

7 ? f⇒=

f(x*)

x*x

f’(x*) f ' x*( ) f x(x*(

---------=

⇒ x x*=

f x( ) f x*( ) x x*–( )f ' x*( ) ……+ +=

⇒ x x* f x*( )f ' x*( )---------------–= ⇒ xk 1+ g xk( )=

y=x

x x1x

y

g

y

x

g' xk( ) 1<

(4-38)

) vector. Iterative

such that

(4-39)

(4-40)

.

(4-41)

, .....

k 1+ Φk≈

aii 0≠

0 i j≥aij i j<

Page 4-21

(3) Iterative Methods

(4-37)+BC in matrix form

where is a (n x n) matrix, and B are (n x 1

methods try to find a matrix and a vector c,

is equivalent to (4-38), then

starting with some initial guess , such that

(a) Point-by-Point Iteration

(I) Jacobi Iteration

AΦ b= or AΦ b– 0=

A Φ

B

Φ BΦ c+=

Φk 1+ BΦk c+= k Iteration count = 1, 2, 3=

Φ1 Φ

AΦ b= rewrite L D U+ +( )Φ b= where

lij0 i j≤aij i j>

= dij

0 i j≠aij i j=

= uij

=

(4-42)

(4-43)

U)Φ D1–

b+

i

2k x3

k 12+– )

0 x3k 12+– )

x2k 0 12+ + )

Page 4-22

or

Accordingly,

where

e.g. 1,

DΦ L U+( )Φ– b+= ⇒ Φ D1–

L +(–=

Φk 1+D

1–L U+( )Φk

– D1–b+=

Φk 1+BΦk

c+=

bij

aijaii------– i j≠

0 i j=

= cij

biaii------= for all

10x1 x2 x3+ + 12= ⇒ x1k 1+ 1

10------ 0 x–(=

x1 10x2 x3+ + 12= ⇒ x2k 1+ 1

10------ x1

k +(=

x1 x2 10x3+ + 12= ⇒ x3k 1+ 1

10------ x– 1

k –(=

(4-44)

(4-45)

(4-46)

ach (Parabolic) ==>

dy

ui j 1+,

0= β ∆x∆y------=

1 j, β2ui j 1+,k+ ]

Page 4-23

e.g. 2, Recall (4-37)

If , then (4-44) becomes

Also recall

If FTCS + + , gives

Discussions:

• Steady problems (Elliptic) can use unsteady approall schemes for parabolic can be used, e.g., ADI

• I.G. but not I.C. ==> No physical meaning except stea

• Larger time-step size.

β2ui j 1–, ui 1– j, 2 1 β2+( )ui j,– ui 1 j,+ β2+++

Jacobi ⇒ ui j,k 1+ 1

2 1 β2+( )----------------------- β2ui j 1–,

k ui 1– j,k ui +

k+ +[=

∆x ∆y= β⇒ 1=( )

ui j,k 1+ 1

4--- ui j 1–,

k ui 1– j,k ui 1+ j,

k ui j 1–,k++ +[ ]=

t∂∂u

x2

2

∂∂ u

y2

2

∂∂ u

+=

∆x ∆y=∆t

∆x( )2--------------

14---=

ui j,n 1+ 1

4--- ui j 1–,

n ui 1– j,n ui 1+ j,

n ui j 1+,n++ +[ ]=

(4-47)

(4-48)

Φk b+

k 1+ BΦk c+=

k b+

k b+ ]

aijΦjk

j i>∑ bi+

----------------------------------

Page 4-24

(II) Gauss-Seidel Iteration

or

e.g.,

AΦ b= ⇒ L D U+ +( )Φ b=

⇒ L D+( )Φ UΦ– b+= ⇒ L D+( )Φk 1+ U–=

⇒ Φk 1+ L D+( )1–

UΦk– L D+( )1–

b+= ⇒ Φ

DΦk 1+ LΦk 1+ UΦ––=

Φk 1+ D1–

LΦk 1+ UΦ––[=

i.e., Φik 1+

aijΦjk 1+

j i<∑ ––

aii-------------------------------------------=

X

i-1 i i+1

j+1

j

j-1 X

x1k 1+ 1

10------ 0 x2

k– x3k 12+–( )=

x2k 1+ 1

10------ x1

k 0 x3k 12+–+( )=

x3k 1+ 1

10------ x– 1

k x2k 0 12+ +–( )=

UL

(III) Convergence Test

uity equation.

teed if

(4-49)

for all i

for all i

x

Page 4-25

(a) Absolute Criterion:

(b) Relative Criterion:

(c) Alternative Ways of Testing:

e.g.1,

e.g.2,

e.g.3, Based on physics, such as contin

(d) Convergence of GS Iteration is guaran

i.e., Diagonally Dominant!!!

Φik 1+ Φi

k– ε1<

Φik 1+ Φi

k–

Φik 1+

----------------------------- ε2<

Φik 1+ Φi

k– ε3<∑

Φik 1+ Φi

k–

Φik 1+

----------------------------- ε4<all i∑

aii aij

j 1=

i j≠

jmax

∑> for i = 1, 2, 3, 4, ......., ima

(4-50)

(4-51)

(4-52)

L D U+ +( )Φk– bk+

k

Rk

k ∞→lim 0⇒

L D U+ +( )Φk– bk+=

Page 4-26

(IV) Delta (∆) Form

Recall

Define residual at the kth iteration as

(a) Jacobi Iterative Method

gives

Convergence requires

(b) Gauss-Seidel Iterative Method

gives

AΦ b= or L D U+ +[ ]Φ b=

Rk

Rk AΦk bk– L D U+ +[ ]Φ k bk–= =

DΦk 1+ L U+( )Φk– bk+= or D Φk 1+ Φk–( ) =

D∆Φk Rk–= where ∆Φk Φk 1+ Φ–=

∆Φk

k ∞→lim 0⇒ and

D L+( )Φk 1+ UΦk– bk+= or D L+( ) Φk 1+ Φk–( )

D L+( )∆Φk Rk–=

(4-53)

a direct method.

have

(4-54)

e, or conditioning

is easily solvable.

(4-55)

(4-56)

Page 4-27

(V) Convergence Analysis of Iterative SchemesAny iterative method is said to be convergent if

in which is the exact solution of by

For an arbitrary splitting of , say , we

where is called the iteration, convergenc

matrix. Matrix should be selected such that it

For a direct method, we also have

(4-54) - (4-55) gives

or

ekk ∞→lim 0⇒ where ek Φk Φ–=

Φ AΦ b=

A A P Q+=

PΦk 1+ QΦk b+–=

P

P

P Q+( )Φ b= or PΦ QΦ b+–=

Pek 1+ Qek–=

(4-57)

trix.

(4-58)

ined by the modulus

(4-59)

…

P1–

Ake1

Page 4-28

or

matrix is called iteration or amplification ma

Convergence requires1

where is the spectral radius of G and is defof the largest eigenvalue ( ) of G, i.e.,

1. See “DIGRESSION” on next page

ek 1+ P1–

Q– ek P

1–Q–

P1–

Q– ek 1–= = =

⇒

ek 1+ P1–

Q–

ke1 P

1–A P–( )–

ke1 I –= = =

ek 1+ Gke1= where G I P

1–A–=

G

ρ G( ) 1≤ or λJ G( ) 1≤ for all J

ρ G( )λ G( )

ρ G( ) Max λJ G( )= for all J

rs, say , i.e., the

sional vector space

vJ

tion of vJ

k 1+ CJλJkvJ

J 1=

n

∑=

Page 4-29

DIGRESSION: Matrix Algebra

If matrix is real and symmetric, then

(a) has n linearly independent eigenvecto

eigenvalues ( ) of are all distinct.

(b) can be used as a basis for the n-dimen

and the error vector .

e.g.,

(c) Also

(d) Since

A(n x n)

A

λJ A

vJ

e1

e1 CJvJ

J 1=

n

∑= i.e., linear combina

e2 Ae1 A CJvJ

J 1=

n

∑ CJAvJ

J 1=

n

∑= = =

AvJ λJvJ= ⇒ e2 CJλJvJ

J 1=

n

∑= ⇒ e

which is easily invertible)

ions can be defined

(4-60)

(4-61)

) I D1–

A–=

D L+ ) 1–A

Page 4-30

Discussion:

(a) If (Recalled: Look for

(b) For Jacobi:

For Gauss-Seidel:

It can be shown that

if is diagonally dominant

the convergence is guaranteed!

(c) Estimation of the convergence rate

The avg rate of error reduction over k iteratby

From (4-57), we have , gives

P A= ⇒ direct P

G D1–

L U+( )– D1–

A D–(–= =

G D L+( ) 1–U– I (–= =

A

∆ekek

e0----------

1k---

= where Norm≡

ek Gke0=

∆ek Gk

1k---

≤

rix Iterative Analysis.”)

(4-62)

te.rder of magnitude

od

s

mptotically for large k only

hod which dampshigh-frequency

damps poorly therors.

rve.

| | | | | |

i-1 i i+1

x

L

λmin = 2∆x

Page 4-31

Asymptotically it can be shown (Very, 1962, “Mat

The logarithm of S is the avg convergence rae.g., For an error to be reduced to an o

(factor 10) in k iterations

(d) Typical residual history of a relaxation meth

S Gkk ∞→lim

1k---

ρ G( )= = ρ Spectral radiu≡

110------

1k---

S≤ ρ G( )= or k1–

log10 ρ G( )[ ]-------------------------------≥ Valid asy

(a) curve: typical for a metrapidly the components but Low-frequency er

(b) curve: Opposite of (a) cu(b)

(a)

k

|R|

| | | | | | | | | | |

0

λmax = L

∆x

e-Over-Relaxation)

inated problem.

e for linear cases for nonlinear

(4-63)

(4-64)

Φk

n

n

Page 4-32

(VI) GS Iteration with Relaxation (SOR, Successiv

• For highly nonlinear and/or convective dom

• Purpose to increase the convergence ratand to restrict the changes of cases.

or

⇒∆

Φi j,k 1+ Φi j,

k ω Φi j,k 1+ Φi j,

k–[ ]+=

From GS Iteration

Relaxation Factor

Φi j,k 1+ ωΦi j,

k 1+ 1 ω–( )Φi j,k+=

The relaxation parameter ω can be

ω

0 ω 1< < ⇒ Under-Relaxatio

1 ⇒ Gauss-Seidel

1 ω 2< < ⇒ Over-Relaxatio

ω 2≥ ⇒ Diverge ??

=

product of its eigen-

)D ωU– ]

er Matrix

1U

0 N = matrix size

R ω( ) 1 ω–( )N=

2<

Page 4-33

Why >

Recall GS in Delta form

SOR

The amplification Matrix is

The determinant is

Also, the determinant of a matrix is equal to the values (similarity transformation)

The Spectral radius will be lower than one if

0 ω 2< <

D L+( )∆Φk Rk

–=

⇒ D ωL+( )∆Φk ωRk

–=

GSOR ω( ) I ω D ωL+( )1–

A– D ωL+( )1–

1 ω–([= =

Lower Matrix Upp

det GSOR ω( ) detI ωD

1–L–

1–det

1 ω–( )I ωD–

–•=

1 0 (1-ω)N

∴ The spectral radius ρ GSOR ω( )[ ]N det GSO≥

1 ω–( ) ρ GSOR ω( )( ) 1<≤ or 0 ω<

(4-65)

(4-66)

ary information

1 sp+

1 sp+

verticalt, ilast

x x x xE

i

Page 4-34

(b) Line-by-Line IterationRecall (Jacob, Gauss-Seidel, SOR, ...)

Rearranging

Main Advantage: Faster transmission of bound

==> Faster Convergent Rate

apΦp aEΦE aWΦW aSΦS aNΦN sp+ + + +=

aSΦSk 1+– apΦP

k 1+ aNΦNk 1+–+ aEΦE

k aWΦWk ++=

or

aWΦWk 1+– apΦP

k 1+ aEΦEk 1+–+ aNΦN

k aSΦSk ++=

W P E

S

N

yes chosen noSweeping Direction => Horizontali.e., for each i, do j = jst, jlast

x

x

x

x

x

xxx

i

j

noChosenline

yes

Sweeping Direction =>i.e., for each j, do i = is

x x x xW P

S

N

j

the boundaryain.

+ Relaxation

Page 4-35

To further increase the convergence rate,

Do

Which one first? ==> Based on Physics

i.e., Which sweep first will transmitinformation faster into the calculation dom

e.g.1,

e.g.2, Engine Problems

Or+ + Relaxation +

(4-67)

0=

t const=

t∂∂u

ax∂∂u

+

stant on

dt⁄ a=

x

u

at

Page 4-36

4-3 Hyperbolic PDEs

e.g. 1, Linear

e.g. 2, Nonlinear

(1) Linear

Consider

t∂∂u

ax∂∂u

+ 0=

t∂∂u

ux∂∂u

+ 0= ort∂∂u

x∂∂ u2

2----- +

t∂∂u

ax∂∂u

+ 0= a 0 Characteristic⇒> dtdx------

1a--- x a–⇒=

Digression:

if u u x t,( ) du⇒t∂∂u

dtx∂∂u

dxdudt------⇒+

t∂∂u dx

dt------

x∂∂u

+dudt------⇒= = = =

4 67–( ) dudt------⇒ 0 du⇒ 0= = along

dxdt------ a

i.e., u remains con

characteristic dx⇒=

The general solution is u f x at–( )=e.g.,

x

u

(4-68)

(4-69)

(4-70)

(4-71)

, i.e.,

∆x)].

∆x( )2].

∆x)]

a∆t∆x---------

1≤

a=

Page 4-37

(a) Explicit Methods(I) Euler’s FTFS (a > 0)

(II) Euler’s FTCS (a > 0)

(III) Euler’s FTBS (1st-order Upwind Scheme)

Discussions:

•Rewrite (4-70) as

Recall u remains constant on the characteristic

uin 1+ ui

n–

∆t-------------------------- a

ui 1+n ui

n–

∆x-------------------------–=

O ∆t( ) (,[

U. U

uin 1+ ui

n–

∆t-------------------------- a

ui 1+n ui 1–

n–

2 ∆x( )--------------------------------–=

O ∆t( ),[

U. U

uin 1+ ui

n–

∆t-------------------------- a

uin ui 1–

n–

∆x-------------------------–=

O ∆t( ) (,[

C. S. C =

uin 1+ ui

n a∆t∆x--------- ui

n ui 1–n–( )–=

dxdt------

ation of du/dt = 0 alongolation between i and i-

x

i-1 i

Page 4-38

Accordingly, (4-71) can be interpreted as an integrdx/dt = a by finding u( P+) through a linear interp1 at level n, and writing

For a fixed (Say)

tddu∫ 0= along

dxdt------ a=

dx/dt=a

i-1 i i+1a∆t

∆t

(n+1)∆t

n∆t

tP

P+

∆x

upn 1+ up+

n uin a∆t

∆x--------- ui

n ui 1–n–( )–= =

∆x xi xi 1––=

a∆t ∆x≤ ora∆t∆x--------- 1≤ i.e., ∆t

∆xa

------≤

Ifa∆t∆x--------- 1= ⇒ Exact Solution

(4-72)

, the generale can be written as

(4-73)

nd a

1-----

Page 4-39

•If a < 0, then FTFS, i.e., (4-68)

•General Expression:

•Using the relationship between equation for the first-order upwind schem

(IV) Lax Method

Recall FTCS

uin 1+ ui

n–

∆t-------------------------- a+

uin ui 1–

n–

∆x------------------------- a–

ui 1+n ui

n–

∆x-------------------------––=

where

a+ 12--- a a+( )= 0≥ and a– 1

2--- a a–( )= 0≤

a± a a, ,

uin 1+ ui

n–

∆t-------------------------- a

ui 1+n ui 1–

n–

2 ∆x( )--------------------------------–

∆x2

------ aui 1+

n 2uin ui –

n+–

∆x( )2------------------------------------------+=

CS forx∂∂u

CS forx2

2

∂∂ u

(4-74)

arting; 2nd-order in

(4-75)

(4-76)

. S. if C 1≤

Page 4-40

==> Lax Method

(V) Midpoint Leapfrog Method: CTCS (not self-stt and x; C. S., C 1)

(VI) Lax-Wendroff Scheme

TSE in time for u

Recall GE

uin 1+ ui

n a∆t2 ∆x( )-------------- ui 1+

n ui 1–n–( )–=

Replaced by ui 1+n ui 1–

n+

2--------------------------------

uin 1+ 1

2--- ui 1+

n ui 1–n+( ) C

2---- ui 1+

n ui 1–n–( )–= C

a∆t∆x---------= C

≤

uin 1+ ui

n ∆t1!-----

t∂∂u

i

n ∆t( )22!

-------------t2

2

∂∂ u

i

n

O ∆t( )3[ ]+ + +=

t∂∂u

ax∂∂

u( )–=

(4-77)

(4-78)

(4-79)

a2

x2

2

∂∂ u

3]

ui 1–n

-------------

2uin ui 1–

n+

∆x)2---------------------------

Page 4-41

(4-76)+(4-77) into (4-75), gives

Then CS ==>

Discussions:

•

•C. S.,

•Recall 1st-order upwind scheme (4-73)

⇒t2

2

∂∂ u

at∂∂

x∂∂u – a

x∂∂

t∂∂u – a

x∂∂

ax∂∂u

– –= = = =

uin 1+ ui

n ∆t( ) ax∂∂u

–

i

n ∆t( )22

------------- a2

x2

2

∂∂ u

i

n

O ∆t( )[+ + +=

uin 1+ ui

n a ∆t( )–ui 1+

n ui 1–n–

2∆x-------------------------------- a2 ∆t( )2

2-------------------

ui 1+n 2ui

n +–

∆x( )2----------------------------------

+=

O ∆t( )2 ∆x( )2,[ ]

C 1≤

uin 1+ ui

n a ∆t( )ui 1+

n ui 1–n–

2 ∆x( )--------------------------------

–a ∆x( ) ∆t( )

2----------------------------

ui 1+n –

(--------------------

+=

(4-80)

e 1st-order upwind

(4-81)

(4-82)

Page 4-42

both schemes can be considered as

•Both schemes can also be written as

where f* is called the numerical flux. For thscheme

and for the Lax-Wendroff scheme

⇒

FTCS Coeffx2

2

∂∂ u

+

uin 1+ ui

n ∆t∆x------ f

i12---+

* fi

12---–

*–

–=

fi

12---+

* 12--- a ui 1+

n uin+( ) a– ui 1+

n uin–( )=

∆ui

fi

12---+

* 12--- a ui 1+

n uin+( ) ∆t

∆x------a2– ui 1+

n uin–( )=

∆ui

(4-83)

(4-84)

ms)

∆t( ) ∆x( )2,[ ]

U. S.

Tridiagonal

Physics

∆t)2 ∆x( )2, ]

U. S.

idiagonal

Page 4-43

(b) Implicit Methods(I) Euler’s BTCS

(II) Crank-Nicolson Scheme

(III) Splitting Methods (Multidimensional Proble

AF

ADI

FS, ...

uin 1+ ui

n–

∆t-------------------------- a

ui 1+n 1+ ui 1–

n 1+–

2∆x----------------------------------–=

12---

a∆t∆x---------

ui 1–n 1+ ui

n 1+–12---

a∆t∆x---------

ui 1+n 1+– ui

n–=

O

uin 1+ ui

n–

∆t--------------------------

12---

t∂∂u

i

n

t∂∂u

i

n 1++ O ∆t( )2[ ]+=

ax∂∂u –

i

na

x∂∂u –

i

n 1+

uin 1+ ui

n–

∆t--------------------------

a2---

ui 1+n ui 1–

n–

2∆x--------------------------------

ui 1+n 1+ ui 1–

n 1+–

2∆x----------------------------------+–=

O ([

Tr

ethods)

2 Versions)

d as predictor andtor)

(4-85)

(4-86)

Page 4-44

(c) Multi-Step Methods (Predictor-Corrector M

Better for nonlinear equations.

(I) Richtmyer/Lax-Wendroff Multi-Step Method (

(i) Richtmyer Formulation (use Lax MethoBTCS as correc

Predictor (Lax Method at )

Corrector (BTCS)

O ∆t( )2 ∆x( )2,[ ]

ui

n12---+

ui

n12---+ 1

2--- ui 1+

n ui 1–n+( )–

∆t 2⁄( )------------------------------------------------------------- a

ui 1+n ui 1–

n–

2∆x--------------------------------

–=

uin 1+ ui

n–

∆t-------------------------- a

ui 1+

n12---+

ui 1–

n12---+

–

2∆x-----------------------------------

–=

C. S.a∆t∆x--------- 2≤

d as predictor andtor)

(4-87)

(4-88)

)

1 uin–

∆x---------------

Page 4-45

(ii) Lax-Wendroff Formulation (use Lax MethoBTCS as correc

Predictor (Lax Method at )

Corrector (BTCS)

(II) MacCormack Multi-Step Method (FTFS/BTBS

ui

12---+

n12---+

ui

12---+

n12---+ 1

2--- ui 1+

n uin+( )–

∆t 2⁄( )------------------------------------------------------- a

ui +n

----------

–=

i-1 i i+1

n

n+1

XXXX

uin 1+ ui

n–

∆t-------------------------- a

ui

12---+

n12---+

ui

12---–

n12---+

–

∆x---------------------------------

–=

C. S.a∆t∆x--------- 1≤

O ∆t( )2 ∆x( )2,[ ]

(4-89)

(4-90)

1 uin– )

in ui

*+ )

Page 4-46

Predictor: FTFS

Corrector: BTBS

Discussions:

•C. S.

•Can also be

ui* ui

n–

∆t------------------ a

ui 1+n ui

n–

∆x--------------------------

–= i.e., ui* ui

n a∆t∆x--------- ui +

n(–=

uin 1+ ui

**–

∆t 2⁄( )----------------------------- a

ui* ui 1–

*–

∆x--------------------------

–= where ui** 1

2--- u(=

⇒ uin 1+ 1

2--- ui

n ui*+( ) a∆t

∆x--------- ui

* ui 1–*–( )–=

a∆t∆x--------- 1≤

FTBSBTFSFTFSBTBS::

n -> n+1

n+1 -> n+2

(4-91)

(4-92)

uin ui 1–

n– )

Page 4-47

•Good for nonlinear problems

•If linear = Lax-Wendroff method

•Revisit:

Rewrite (4-90) as

From (4-89)

(4-92) into (4-91) gives Lax-Wendroff Scheme

uin 1+ ui

n 12--- ui

* u– in( ) a∆t

2∆x---------- ui

* ui 1–*–( )–+=

uin 1+ ui

n ∆t2-----

ui* u– i

n( )

∆t-----------------------

a∆x------ ui

* ui 1–*–( )–+=

uin 1+ ui

n ∆t2-----

t∂∂u a

x∂∂u

*–+=

uin 1+ ui

n ∆t2-----

t∂∂u

FT t∂∂u

BS

*++=

ui* ui

n a∆t∆x--------- ui 1+

n uin–( )–= and ui 1–

* ui 1–n a∆t

∆x---------(–=

ui* ui 1–

*– uin ui 1–

n a∆t∆x--------- ui 1+

n 2uin– ui 1–

n+( )––=

)

BC

u 0 t,( ) 0=

u 300 t,( ) 0=

1

100

220

00

Page 4-48

Example 1:

Example 2:

t∂∂u

ax∂∂u

+ 0= a 300 m/s ( sound speed at altitude of 10 km≈=

I.C.

u x 0,( ) 0= 0 x 50≤ ≤

u x 0,( ) 100 π x 50–60

-------------- sin

= 50 x 110≤ ≤

u x 0,( ) 0= 100 x 300≤ ≤

t2

2

∂∂ u

a2

x2

2

∂∂ u

– 0= with BC Example≈

I.C.u x 0,( )∂

t∂------------------- 0= and

u x 0,( ) 0= 0 x≤ ≤

u x 0,( ) 100 π x 100–120

----------------- sin

= 100 x≤ ≤

u x 0,( ) 0= 220 x 3≤ ≤

e Scheme1)

ion error

me

t t = n∆t

mes are developed; and (2) the

f E between n & n+1

Page 4-49

(d) Explicit Schemes Revisited

Recall 1st-order upwind scheme

•No oscillations (GOOD!) => monotone variation (Monoton

•Smear out the flow variables (BAD!) => diffusive, dissipat

•Can be considered as a Godunov-Type Scheme

(I) Basic Godunov Scheme•There are three steps involved in the basic Godunov Sche

(1) Define a piecewise constant approximation of u a

(2) Obtain local exact solution at the cell interfaces

(3) Avg the state variables after a time interval ∆t

1. A monotonic scheme is defined by: (1) as the solution proceeds in time, no new local extrelocal minimum is non-decreasing whereas the local maximum is non-increasing.

t∂∂u

x∂∂E

+ 0= a x b≤ ≤ u u x t,( )=

tdd

u x t,( ) xda

b

∫ E b t,( ) E a t,( )–[ ]–=

un 1+ x( ) xdb

∫ un x( ) xdb

∫– ∆t E b( ) E a( )–[ ]–= E time avg o≡

) ) )

u x t,( ) xdi 1 2⁄–( )

i 1 2⁄+( )

∫

a∆t)uin

xd

x x xi-1 i i+1

x

t = (n+1)∆t

Step 3

Page 4-50

e.g. the 1st-order upwind scheme

•Monotone schemes are only first-order??

•How about 2nd-order upwind (dispersion error?).

Define the avg state variable over the cell (i-1/2, i+1/2) as ui1∆x------=

uin 1+ ui

n ∆t∆x------ E ui 1 2⁄+

n( ) E ui 1 2⁄–n( )––= ) )

uin 1+ ui

n a∆t∆x--------- ui

n ui 1–n–( )–

1∆x------ a∆t( )ui 1–

n ∆x –(+= =

=1∆x------ un x a∆t–( )

i 1 2⁄–( )

i 1 2⁄+( )

∫

x x x x x xi-1 i i+1 i-1 i i+1

x

u

x

u ut = n∆t

Step 1 Step 2

a∆t

a∆t

∆x-a∆t

(4-93)pstream-centeredpstream-centeredpstream-centeredpstream-centered

for Conservation for Conservation for Conservation for Conservation, MUSCL), MUSCL), MUSCL), MUSCL)

xxxx

Page 4-51

(II) Second-Order Upwind Scheme (a>0)

i.e.,

givesx∂∂u

i

n 3uin 4ui 1–

n ui 2–n+–

2∆x---------------------------------------------------=

uin 1+ ui

n a∆t2∆x---------- 3ui

n 4ui 1–n ui 2–

n+–( )–=

x x xi-1 i i+1

x

u

=> i.e., piecewise linear within each cell

cell avg + slope (cell avg + slope (cell avg + slope (cell avg + slope (Monotone UMonotone UMonotone UMonotone U

Scheme Scheme Scheme Scheme Laws Laws Laws Laws

n∆t

(n+1)∆t

uin

uin 1+

x i-1 i-1 i-1 i-1

uuuu

x i i i i

x i+1 i+1 i+1 i+1

e

iscontinuity

(4-94)

order upwind

Scheme

Page 4-52

(III) Total-Variation-Diminshing (TVD) Schem•Two types: upwind and symmetric

•Most of them 2nd-order everywhere but 1st-order near d

Recall (4-80), (4-81), and (4-82)

Symmetric TVD Scheme1

Where

1. Yee, H.C., 1987, “Upwind and Symmetric Shock-Capturing Schemes,” NASA TM-89464

uin 1+ ui

n ∆t∆x------ f

i12---+

* fi

12---–

*–

–=

fi

12---+

* 12--- a ui 1+

n uin+( ) a– ui 1+

n uin–( )=

fi

12---+

* 12--- a ui 1+

n uin+( ) ∆t

∆x------a2– ui 1+

n uin–( )=

for 1st-

for L-W

fi

12---+

* 12--- a ui 1+

n uin+( ) a 1 Q

i12---+

–

– ui 1+n ui

n–( )=

(4-95)

ui 2+ ui 1+–

ui 1+ ui–-------------------------------=

1–

12---

u ∆i

32---+

u,

∆i

12---+

u–

Page 4-53

Alternative Form: Define

Qi

12---+

Q ri

12---+

– ri

12---+

+,

= ri

12---+

–ui ui 1––

ui 1+ ui–-----------------------= r

i12---+

+

Q r– r+,( ) minmod 1 r–,( ) minmod 1 r+,( )+=

minmod x y,( ) x( )sgn max 0 min x y x( )sgn⋅,[ ], ⋅=

x if x y and xy 0><y if x y and xy 0>>0 if xy 0<

=

minmod 1 r±,( )min 1 r±,( ) r± 0>

0 r± 0≤

=

∆i 1 2⁄+ u ui 1+ ui–=

fi

12---+

* 12--- a ui 1+

n uin+( ) a ∆

i12---+

u Qi

12---+

–

–=

Qi

12---+

Q ∆i

12---–

u ∆i

12---+

u ∆i

32---+

u, ,

minmod ∆i

12---–

u ∆i

12---+

u,

minmod ∆i +

+= =)

) )

uation

(4-96)

(4-97)Eu2

2-----=

x

x

InitialProfile

compression

Page 4-54

(2) Nonlinear

Consider the nonlinear inviscid Bergers eq

or, in conservation law form

t∂∂u

ux∂∂u

+ 0=

t∂∂u

x∂∂ u2

2----- + 0= or

t∂∂u

x∂∂E

+ 0= where

Digression:

i.e.,

u remains constant on the

characteristic

du 0= alongdxdt------ u=

dxdt------ u=

t

u

expansion

(4-98)

(4-99)

iemann invariant)

(4-100)

ui 1–n

--------------

u∂∂E

u=i

n????

i12---+

n . .??

Page 4-55

(a) Explicit Methods(I) 1st-order Upwind Scheme

Recall linear case

then, for the nonlinear case

or if

then flux propagation speed characteristic variable (R

uin 1+ ui

n–

∆t-------------------------- a

ui 1+n ui 1–

n–

2 ∆x( )--------------------------------–∆x2

------ aui 1+

n 2uin +–

∆x( )2---------------------------------+=

uin 1+ ui

n–

∆t--------------------------

Ei 1+n Ei 1–

n–

2 ∆x( )---------------------------------–

∆x2

------ aui 1+

n 2uin ui 1–

n+–

∆x( )2-----------------------------------------------+= a =

uin 1+ ui

n ∆t∆x------ f

i12---+

* fi

12---–

*–

–=

fi

12---+

* 12--- Ei 1+

n Ein+( ) a– ui 1+

n uin–( )=

should Ei 1+n Ei

n–=

One of Roe‘s approximate Riemann solver conditions

au∂∂E

u= =

(4-101)

(4-102)

(4-103)

(4-104)

Ax∂∂E

3]

Page 4-56

(II) Lax-Wendroff Method

TSE in time for u

Recall GE

where

(4-102)+(4-103) into (4-101), gives

uin 1+ ui

n ∆t1!-----

t∂∂u

i

n ∆t( )22!

-------------t2

2

∂∂ u

i

n

O ∆t( )3[ ]+ + +=

t∂∂u

x∂∂E

–=

⇒t2

2

∂∂ u

t∂∂

x∂∂E –

x∂∂

t∂∂E –

x∂∂

u∂∂E

t∂∂u

–

x∂∂

= = = =

Chain Rule Ax∂∂E

–

Au∂∂E

u∂∂ u2

2----- u= = =

uin 1+ ui

n ∆t( )x∂∂E

–

i

n ∆t( )22

-------------x∂∂

Ax∂∂E

i

n

O ∆t( )[+ + +=

i+3

i12---–

nEi

n Ei 1–n–

∆x--------------------------

---------------------------------------------

Ain+ ) Ei

n Ei 1–n–( )

-----------------------------------------------

Page 4-57

Then CS ==>

Discussions:

•2nd-order method => dispersive

•C. S.,

i-3 i-2 i-1 i i+1 i+2i-1/2 i+1/2

∆x

x∂∂E

i

n Ei 1+n Ei 1–

n–

2∆x---------------------------------=

x∂∂

Ax∂∂E

i

nA

x∂∂E

i12---+

nA

x∂∂E

i12---–

n–

∆x----------------------------------------------------------

Ai

12---+

nEi 1+

n Ein–

∆x--------------------------

A–

∆x------------------------------------------------------= =

⇒x∂∂

Ax∂∂E

i

n12--- Ai 1+

n Ain+( ) Ei 1+

n Ein–( ) 1

2--- Ai 1–

n(–

∆x( )2---------------------------------------------------------------------------------------------=

∆t∆x------ umax 1≤

(4-105)

(4-106)

esult.

cDonald + CN Scheme

(4-107)

(4-108)

Page 4-58

(III) MacCormack Method

Predictor

Corrector

Discussions:

•2nd-order method => dispersive

•C. S., . Local gives better r

(b) Implicit Method - Beam and Warming / Briley and M

CN scheme

Since

ui* ui

n ∆t∆x------– Ei 1+

n Ein–( )=

uin 1+ 1

2--- ui

n ui*+( ) ∆t

∆x------– Ei

* Ei 1–*–( )=

∆t∆x------ umax 1≤ C 1≈

uin 1+ ui

n ∆t2-----

t∂∂u

i

n 1+

t∂∂u

i

n+ O ∆t( )3[ ]+ +=

t∂∂u

x∂∂E

–=

(4-109)

(4-110)

(4-111)

(4-112)in

Page 4-59

(4-108) into (4-107) gives

Linearization:

(4-110) into (4-109)

then CS

uin 1+ ui

n ∆t2-----

x∂∂E

–

i

n 1+

x∂∂E

–

i

n+ O ∆t( )3[ ]+ +=

Ein 1+ Ei

nt∂∂E

i

n∆t O ∆t( )2[ ]+ +=

Ein

u∂∂E

i

n

t∂∂u

i

n∆t O ∆t( )2[ ]+ +=

Ein Ai

nui

n 1+ uin–

∆t-------------------------- O ∆t( )+ ∆t O ∆t( )2[ ]+ +=

Ein Ai

n uin 1+ ui

n–( ) O ∆t( )2[ ]+ +=

uin 1+ ui

n ∆t2-----– 2

x∂∂E

i

n

x∂∂

Ain ui

n 1+ uin–( )[ ]+=

∆t4∆x---------- Ai 1–

n ui 1–n 1+– ui

n 1+ ∆t4∆x---------- Ai 1+

n ui 1+n 1++ + RHS=

., > 2nd-order

(4-113)

(4-114)

equations): LHS

(4-115)

i 1+n ∆t

4∆x---------- Ai 1–

n ui 1–n–

2 ) εe18---≈

εe

Page 4-60

where

Discussions:

•U. S.

•2nd-order method => dispersive

•Linear Damping (should be higher order, i.e

(i) Explicit: RHS

For multidimensional problems

(ii) Implicit (maintain tridiagonal system of

RHSin ui

n 12---∆t∆x------ Ei 1+

n Ei 1–n–( )–

∆t4∆x---------- Ai 1+

n u+=

RHSinew RHSi

n De+=

De εe ∆x( )4x4

4

∂∂ u

i

n

– εe ui 2–n 4ui 1–

n– 6uin 4ui 1+

n– ui +n+ +(–= =

De εe ∆x( )4x4

4

∂∂ u

∆y( )4y4

4

∂∂ u

…+ +i

n

–=

Di εi ∆x( )2x2

2

∂∂ u

∆y( )2y2

2

∂∂ u

…+ +i

n

–= εi 2≅

(4-116)

(4-117)

(4-118)

Ain ui

n 1+ uin–( )[ ]

1–------

1 Ei 1–n– )

Page 4-61

•Delta ( ) Form:

Define

then from (4-111)

gives

CS gives

or

Examples:

∆

∆uin ui

n 1+ uin–=

uin 1+ ui

n ∆t2-----– 2

x∂∂E

i

n

x∂∂

+=

∆uin ∆t

2-----

x∂∂

Ain∆ui

n( )+ ∆t( )–x∂∂E

i

n=

∆uin ∆t

2-----

Ai 1+n ∆ui 1+

n Ai 1–n ∆ui 1–

n–

2∆x--------------------------------------------------------------------

+ ∆tEi 1+

n Ein–

2∆x---------------------------

–=

∆t4∆x---------- Ai 1–

n ∆ui 1–n– ∆ui

n ∆t4∆x---------- Ai 1+

n ∆ui 1+n+ +

∆t2∆x---------- Ei +

n(–=

Plus Damping Terms u ∆u→

(4-119)

x

xu1t u2t

t

nsion

0

Page 4-62

(c) Symmetric TVD Scheme

Define

x

u

x

t t

u

t = 0 t > ts

xs

ts

t = 0

u1

u2

Compression Expa

ai

12---+

∆i 1 2⁄+ E

∆i 1 2⁄+ u----------------------- ∆i 1 2⁄+ u 0≠

u∂∂E

ui∆i 1 2⁄+ u =

=

(4-120)

(4-121)

ui)

12---+

0.125≤

ai

32---+∆

i32---+

u

ai

12---+∆

i12---+

u–

Page 4-63

or

then

where

ai

12---+ u∂

∂Eu

i12---+

ui

12---+

12--- ui 1+ +(==

fi

12---+

* 12--- Ei 1+

n Ein+( ) a

i12---+∆

i12---+

u Qi

–

–=

Ψ ai

12---+

)

Ψ z( )z z ε≥

z2 ε2+2ε

----------------- z ε< where 0 ε≤

=

Qi

12---+

Q ai

12---–∆

i12---–

u ai

12---+∆

i12---+

u ai

32---+∆

i32---+

u, ,

=

minmod ai

12---–∆

i12---–

u ai

12---+∆

i12---+

u,

minmod ai

12---+∆

i12---+

u,

+=

) )

(TV)?

(4-122)

(4-123)

governing by the

(4-124)

,

(4-125)

e TVD1.

mes.

1 uin–

Page 4-64

What is TVD? What is Total Variation

In discrete form

An important property of physical solutionsfollowing equation

is that TV(u) does not increase with time, i.e.

A numerical scheme satisfied (4-125) is said to b

(d) Essentially Non-Oscillatory (ENO) Schemes

1. All monotone schemes, such as the 1st-order upwind scheme, are first-order TVD sche

⇒

TV u( )x∂∂u xd

Ω∫=

TV u( )ui 1+ ui–

xi 1+ xi–----------------------- xi 1+ xi–( )

i∑ ui 1+ ui–

i∑= =

t∂∂u

x∂∂f

+ 0=

TV un 1+( ) TV un( )≤ or ui 1+n 1+ ui

n 1+–

i∑ ui +

n

i∑≤

(4-126)

(4-127)

ρw

ρuw

ρvw

w2

p+

et p+ )w

Page 4-65

(3) Euler Equations (Inviscid Flow)

Recall

• Quasi-One-Dimensional Euler Equations:

t∂∂Q

x∂∂E

y∂∂F

z∂∂G

+ + +x∂

∂Evy∂

∂Fvz∂

∂GvS+ + +=

Q

ρρu

ρv

ρw

ρet

= E

ρu

ρu2

p+

ρuv

ρuw

ρet p+( )u

= F

ρv

ρuv

ρv2

p+

ρvw

ρet p+( )v

= G

ρρ(

=

et e12--- u

2v2

w2

+ +( ) .........+ +=

ℜ ℜ x( )= andt∂∂ ℜ Q( )

x∂∂E H–+ 0=

Q

ρρu

ρet

= E ℜρu

ρu2

p+

ρet p+( )u

= Hdℜdx--------

0

p

0

=

(4-128)

(4-129)

(4-130)

1Q

n–

t)2]

O ∆t( )2[ ]

O ∆t( )2[ ]+

q2

e1

q2

e2

q2

e3

Page 4-66

Backward Time:

Linearization

or

ℜ ∆Q∆t--------

x∂∂E

n 1+H

n 1+–+ 0= where ∆Q Q

n +=

En 1+

En

t∂∂E

n∆t O ∆t( )2[ ]+ + E

nQ∂∂E

t∂∂Q

ℜ∂∂E ℜ∂

t∂-------+

n∆t O ∆([+ += =

⇒t∂∂E

Q∂∂E

t∂∂Q

Q∂∂E ∆Q

∆t--------≈= ⇒ E

n 1+E

nQ∂∂E ∆Q

∆t--------

n∆t+ +=

En 1+

En

Q∂∂E

n∆Q O ∆t( )2[ ]+ += similarly H

n 1+H

nQ∂∂H

n∆Q+=

Digression:

if E

e1

e2

e3

= and Q

q1

q2

q3

= thenQ∂∂E

q1∂

∂e1q2∂

∂e1∂

∂

q1∂

∂e2q1∂

∂e2∂

∂

q1∂

∂e3q1∂

∂e3∂

∂

=

(4-131)

(4-132)

obian matrix

2a

2 γ pρ---=

a2

γ γ 1–( )-------------------12---u

2+

A

Page 4-67

Define Jacobian matrices and as

and

where a is the speed of sound. Accordingly, Jac

A B

AQ∂∂E ℜ

0 1 0

γ 3–2

----------- u

2 γ 3–( )u– γ 1–( )

γuet– γ 1–( )u3+ γet

3 γ 1–( )2

-------------------u2

– γu

= =

BQ∂∂H γ 1–( ) dℜ

dx--------

0 0 0

12---u

2u– 1

0 0 0

= =

For ideal gas γcpcv-----= p ρe γ 1–( )= e et

12---u–=

⇒ p ρ γ 1–( ) et12---u

2–

= etp

ρ γ 1–( )-------------------12---u

2+= =

(4-133)

atrix

(4-134)

(4-135)

A

yperbolic

1 l2 l3 λ1

Page 4-68

can also be rewritten as

(1) Flux Vector Splitting Scheme(a) Eigenvalues and Eigenvectors of Jacobian M

For = u, the left eigenvector is

A ℜ

0 1 0

γ 3–2

----------- u

2 γ 3–( )u– γ 1–( )

ua2

γ 1–-----------–

12---γ 1– u

3+

a2

γ 1–-----------

32--- γ– u

2+ γu

ℜ= =

det A λ I– 0= ⇒ λ1 u λ2 u a+ λ3 u a–=,=,= h⇒

λ1

l1 l2 l3 λ1

0 1 0

γ 3–2

----------- u

2 γ 3–( )u– γ 1–( )

ua2

γ 1–-----------–

12---γ 1– u

3+

a2

γ 1–-----------

32--- γ– u

2+ γu

u l=

⇒ l1 l2 l3 λ11

u2

2------ γ 1–

a2----------- – u

γ 1–

a2----------- γ 1–( )a2–=

envectors are:

(4-136)

(4-137)

by either

(4-138)

u] β γ 1–( )

)u] β γ 1–( )

1

ρa 2--------------=

2, 3

Page 4-69

Similarly, for and , the respective left eig

Accordingly, the left eigenvector matrix is

The right eigenvector matrix can be obtained

λ2 λ3

For λ2 u a+= l1 l2 l3 λ2β u2

2------ γ 1–( ) ua– β a γ 1–( )–[=

For λ3 u a–= l1 l2 l3 λ3 β u2

2------ γ 1–( ) ua+ β a γ 1–(+[–=

L

1u2

2------ γ 1–

a2----------- – u

γ 1–

a2----------- γ 1–( )a2–

β u2

2------ γ 1–( ) ua– β a γ 1–( )u–[ ] β γ 1–( )

β u2

2------ γ 1–( ) ua+ β a γ 1–( )u+[ ]– β γ 1–( )

= where β

0 1 0

γ 3–3

----------- u

2γ 3–( )u– γ 1–( )

ua2

γ 1–-----------–

12---γ 1– u

3+

a2

γ 1–-----------

32--- γ– u

2+ γu

r1

r2

r3 λ i( )

λ i( )

r1

r2

r3 λ i( )

= for i = 1,

⇒ R

r1

r2

r3 λ1

r1

r2

r3 λ2

r1

r2

r3 λ3

=

(4-139)

(4-140)

rce term

(4-141)

(4-142)

α ρa 2----------=

0=

Page 4-70

or

NOTE: It can be shown that

(b) The Compatibility Equations

Consider the 1D Euler equations without sou

From (4-141)

R L1–

1 α α–u α u a+( ) α u a–( )

12---u

2 α 12---u

2ua

a3

γ 1–-----------+ +

α 12---u

2ua–

a2

γ 1–-----------+

= = where

LAR

λ1 0 0

0 λ2 0

0 0 λ3

Λ Eigenvalue Matrix≡= =

t∂∂Q

x∂∂E

+ 0= ort∂∂Q

Q∂∂E

x∂∂Q

+ 0t∂∂Q A

x∂∂Q

+⇒=

L × ⇒ Lt∂∂

Q( ) LAx∂∂Q

+ 0=

RL× ⇒ Lt∂∂

Q( ) LARLx∂∂

Q( )+ 0=

omes

(4-143)

(4-144)

(4-145)

(4-146)

Variable Vector

A–

λ3

Page 4-71

If is a constant (or locally) then (4-142) bec

(c) Flux Vector Splitting

From (4-128) + (4-129) + (4-130) gives

or

Split as following:

L

t∂∂W Λ

x∂∂W

+ 0= where W LQ Characteristic (Riemann)≡=

ℜ ∆Q∆t--------

x∂∂

EnQ∂∂E

n∆Q+ Hn

Q∂∂H

n∆Q+–+ 0=

ℜ ∆Q∆t--------

x∂∂

A∆Q[ ] B∆Q–+En∂x∂

---------– Hn+=

A

A RΛL R Λ+ Λ–+ L R Λ+L RΛ– L+ A+ += = = =

where Λ± 12--- Λ Λ±( ) 1

2---

λ1 λ1± 0 0

0 λ2 λ2± 0

0 0 λ3 ±

= =

(4-147)

(4-148)

(4-149)

(4-150)

(4-151)

Hn

Ei– )n ∆xHi

n–

Page 4-72

(4-146) into (4-145) gives

or

NOTE: If

(4-149) into (4-148) gives

BS for “+” and FS for “-”

ℜ ∆Q∆t--------

x∂∂

A+ A–+ ∆Q B∆Q–+

En∂x∂

---------– Hn+=

ℜ I ∆tA+∂x∂----------

A–∂x∂----------- B–++

∆Q ∆t En∂x∂

--------- Hn––=

E AQ= then E A+ A–+ Q E+ E –+= =

ℜ I ∆tA+∂x∂----------

A–∂x∂----------- B–++

∆Q ∆t E+∂x∂

---------- n E –∂

x∂------------ n

–+–=

ℜ iI∆t∆x------ Ai

+ Ai 1–+– Ai 1+

– Ai–– Bi–++

∆Q∆t∆x------ Ei

+ Ei 1–+–( )n Ei 1+

– –(+–=

⇒ CLi ∆Qi 1– CMi ∆Qi CNi ∆Qi 1++ + RHSi=

(4-152) Bi∆t–

y

ic Inlet

Outlet

Inlet

Outlet

Page 4-73

where

BCs:

CLi∆t∆x------Ai 1–

+–= CNi∆t∆x------Ai 1–

–= CMi ℜ iI∆t∆x------ Ai

+ Ai––

+=

and RHSi∆t∆x------ Ei

+ Ei 1–+–( )n Ei 1+

– Ei––( )n ∆xHi

n–+–=

Nozzle:

Inflow

Outflow

i+1 i i-1x

t

Boundary

Boundar

ℜ x( )

Case 1 :Supersonic Inlet

Supersonic Outlet Case 2 :

Superson

Subsonic

Case 3 :Subsonic Inlet

Subsonic Outlet Case 4 :

Subsonic

Supersonic

etric Form

(4-153)

(4-154)

l qi

12---+

l–

)

λi

12---+

3

l

λil+ )

Page 4-74

(2) 1st-order Upwind and LW Schemes in Symm

(3) Symmetric TVD Scheme

Consider

where

t∂∂Q

x∂∂E

+ 0=

Qn 1+

Qn ∆t

∆x------ E

i12---+

* Ei

12---–

*–

+=

Ei

12---+

* 12--- E

i 1+Ei R

i12---+Φ

i12---+

–+

= and φi

12---+

l ψ λi

12---+

l

∆i

12---+

w

=

φi

12---+

l Elements of Φi

12---+

, l = 1, 2, 3≡ λi

12---+

l λi

12---+

1⇒ λi

12---+

2

∆i

12---+

wl Elements of Li

12---+∆

i12---+

Q (3x3)(3x1) = (3x1)⇒≡

qi

12---+

l q ∆i

12---–

wl , ∆i

12---+

wl ∆i

32---+

wl,=

minmod ∆i

12---–

wl ∆i

12---+

wl,

minmod ∆i

12---+

wl ∆i

32---+

wl,

∆i

12---+

w–+=

ψ z( ) As before (entropy correction function)≡ and λi

12---+

l 12--- λi 1+

l(=

) )