Esquemas Para Ec. de Convedascción

8/18/2019 Esquemas Para Ec. de Convedascción

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Computational Fluid Dynamics

Wave Equation

Grétar Tryggvason Spring 2013

http://www.nd.edu/~gtryggva/CFD-Course/


! 2

f

! t 2 " c

2 ! 2

f

! x2= 0

First write the equation as a system of first order equations

u =! f

! t ; v =

! f

! x;

! u

! t " c

2 ! v

! x = 0

! v

! t "

! u

! x = 0

yielding

Introduce

from the pde

since !

2 f

! t ! x=

! 2 f

! x! t

Wave equation


! u

! t " c

2 ! v

! x = 0

! v

! t "

! u

! x = 0

! u

! t

! v

! t

"

#

$$$$

%

&

''''

+ 0 (c2

(1 0

"

#$

%

&'

! u

! x

! v

! x

"

#

$$$$

%

&

''''

= 0

det AT !" I( ) = det

!" !1

!c2 !"

#

$%%

&

'((= "

2 ! c2( ) = 0

!" = ±c

! =1

2ab ± b2

" 4ac( )

b = 0; a =1; c =!c2

We can also use

with

To find the characteristics

Wave equation


t

x

P

Two characteristic lines dt

dx= +

1

c; dt

dx= !

1

c

dt

dx= +

1

c

dt

dx= !

1

c

Wave equation


l1

! u

! t " c2

! v

! x= 0

# $ %

& ' (

+l2

! v

! t "

! u

! x = 0

# $ %

& ' (

!" !1

!c2 !"

#

$%%

&

'((

l1

l2

#

$

%%

&

'

((= 0

!" = cTo find the solution we needto find the eigenvectors

Take l1=1

!c l1! l

2= 0 " l

2=!cFor ! = +c

+c l1! l

2= 0 " l

2= +cFor ! ="c

Wave equation


1 ! u

! t " c2

! v

! x = 0

# $ %

& ' ( " c

! v

! t " ! u

! x = 0

# $ %

& ' ( Add the

equations

l1=1 l

2=!cFor ! = +c

For ! ="c

l1=1 l

2= +c

! u

! t + c

! u

! x

"

# $

%

& ' ( c

! v

! t + c

! v

! x

"

# $

%

& ' = 0

du

dt ! c

dv

dt = 0 on

dx

dt = +c

du

dt + c

dv

dt = 0 on

dx

dt = !c

Relationbetween the

total derivative

on thecharacteristic Similarly:

Wave equation


2/8


For constant c

du

dt

! c dv

dt

= 0 on dx

dt

= +c

du

dt + c

dv

dt = 0 on

dx

dt = !c

dr1

dt = 0 on

dx

dt = +c where r

1= u ! cv

dr2

dt = 0 on

dx

dt = !c where r

2= u + cv

r1 and r2 are called the Rieman invariants

Wave equation


The general solution can

therefore be written as:

f x, t ( ) = r1 x ! ct ( ) + r2 x + ct ( )

r1 x( ) =

! f

! t " c

! f

! x

#

$ % &

' ( t =0

r2 x( ) =

! f

! t + c

! f

! x

#

$ % &

' ( t =0

where

Can also be verified by direct substitution

Wave equation


Domain ofInfluence

Domain ofDependence

t

x

P

Two characteristic lines dt

dx= +

1

c; dt

dx= !

1

c

dt

dx= +

1

c

dt

dx= !

1

c

Wave equation-general


Ill-posedproblems


Ill-posed Problems

! 2 f

! t 2 = "

! 2 f

! x2

Consider the initial value problem:

! 2 f

! t 2 +

! 2 f

! x2 = 0

This is simply Laplace’s equation

which has a solution if " f / "t or f are given onthe boundaries


Here, however, this equation appeared as an initialvalue problem, where the only boundary conditionsavailable are at t = 0. Since this is a second order

equation we will need two conditions, which we may

assume are that f and " f / " x are specified at t =0.

Ill-posed Problems


3/8


Look for solutions of the type:

f = ak (t )eikx

d 2a

k

dt 2

= k 2a

k

Substitute into: !

2 f

! t 2 +

! 2 f

! x2 = 0

to get:

Ill-posed Problems

f x,t ( ) = ak (t )eikx

k

!

The general solution can be written as:

where the a’s dependon the initial conditions


)0(;)0( ikaa

d 2a

k

dt 2

=

k

2

ak

determined by initial conditions

General solution

ak (t ) = Ae

kt + Be

!kt

B A,

Therefore: !")(t a as !"t

Ill-posed Problem

Ill-posed Problems

Generally, both A and B are non-zero


Ill-posed Problems

Long wave with short wave perturbations


! f

!t = D

!2 f

! x2; D < 0

has solutions with unbounded growth rate for high wavenumber modes and is therefore an ill-posed problem

Similarly, it can be shown that the diffusionequation with a negative diffusion coefficient

Ill-posed Problems


Ill-posed problems generally appear when the initial orboundary data and the equation type do not match.

Frequently arise because small but important higher ordereffects have been neglected

Ill-posedness generally manifests itself in the exponential

growth of small perturbations so that the solution does not

“depend continuously on the initial data”

Inviscid vortex sheet rollup, multiphase flow models andsome viscoelastic constitutive models are examples ofproblems that exhibit ill-posedness.

Ill-posed Problems


Classical Methodsfor Hyperbolic

Equations


4/8


! 2 f

! t 2 " c 2 !

2 f

! x2= 0

! u

! t " c

2 ! v

! x = 0

! v

! t "

! u

! x = 0

The wave equation:

Write as:

In general:

! u

! t

! v

! t

"

#

$$$$

%

&

''''

+

a11

a12

a21

a22

"

#

$$

%

&

''

! u

! x

! v

! x

"

#

$$$$

%

&

''''

= 0

Most of the issues involved can be addressed by examining:

! f

! t + U

! f

! x= 0

Methods for Advection


Forward in Time,Centered in Space

(FTCS) andUpwind


We will start by examining the linear advection equation:

! f

! t + U

! f

! x= 0

The characteristic for this equation are:

dx

dt =U ;

df

dt = 0;

Showing that theinitial conditions are

simply advected by a

constant velocity U

t

f

f

x



A simple forward in time, centered in spacediscretization yields

! f

! t + U

! f

! x= 0

f j n+1

= f j n! "t

2hU ( f j +1

n! f j !1

n)

j-1 j j+1

n

n+1



This scheme is O(#t, h2) accurate, but a stabilityanalysis shows that the error grows as

! n+1

! n =1" i

U #t

2hsin kh

Since the amplificationfactor has the form 1+i()

the absolute value of this

complex number is alwayslarger than unity and the

method is unconditionallyunstable for this case.

iU !t

2hsinkh

1

! n+1

! n



A simple forward in time but “upwind” in spacediscretization yields

! f

! t + U

! f

! x= 0

f j n+1

= f j n! " t

hU ( f j

n! f j !1

n)

j-1 j

n

n+1 This schemeis O(#t, h)

accurate.

Another scheme for

Flow direction



5/8


To examine the stability we use the von Neuman’s method:

! j

n+1" ! j

n

#t +U

h(! j

n" ! j "1

n) = 0

! j n= !

neikx j

! n+1" !

n

#t +U

! n

h(1" e

" ikh)= 0

The evolution of the error is governed by:

Write the error as:

! n+1

! n

=1"U #t

h(1" e

"ikh)

G =! n+1

! n

=1" # (1" e"ikh

), # =U $t

h

Amplification factor

G =1! " + " e! ikh

Or: G


6/8


Finite Volume point of view:

x j -1/2 x j +1/2

f j !1 f j

x

f

f j +1

f j n+1

= f j n! "t

h(F j +1/ 2

n! F j !1/ 2

n) = f j

n! "t

hU ( f j

n! f j !1

n)

F j +1/ 2=Uf j n

F j !1/ 2=Uf j !1n



Generalized Upwind Scheme (for both U > 0 and U < 0 )

f j n+1

= f j n!U "t

h( f j

n! f j !1

n ), U > 0

f j

n+1= f j

n!U "t

h( f j +1

n! f j

n), U < 0

Define: U +

=1

2U + U ( ), U ! =

1

2U ! U ( )

The two cases can be combined into a single expression:

f j

n+1= f j

n! "t

hU +( f j

n! f j !1

n)+ U

!( f j +1

n! f j

n)[ ]

Or, substituting !+

U U ,

f j n+1

= f j n!U

"t

2h( f j +1

n! f j !1

n)+

U "t

2h( f j +1

n! 2 f j

n+ f j !1

n)

central difference + numerical viscosity Dnum =U h

2


Other First Order Schemes


Implicit (Backward Euler) Method

- Unconditionally stable - 1st order in time, 2nd order in space - Forms a tri-diagonal matrix (Thomas algorithm)

( ) 02

1

1

1

1

1

=!+"

!+

!

+

+

+

n

j

n

j

n

j

n

j f f

h

U

t

f f

n

j

n

j

n

j

n

j f

t f

h

U f

t f

h

U

!="

!+

+

"

++

+

1

2

1

2

1

1

11

1

j

n

j j

n

j j

n

j j C f b f d f a =++

+

!

++

+

1

1

11

1



Lax-Fredrichs method

- stable for $ < 1

- 1st order in time, 1nd order in space - Conditionally consistent

f j n+1

! 1

2 f j +1

n+ f j !1

n( )"t

+U

2h f j +1

n! f j !1

n( ) = 0

( ) xxx xx

f Uh

f Uh

2

2

13

1

2! !

! "+#

$ %

&' ( "

Error term:

! =U "t

h

h

! =

h2

U "t



Second Order Schemes


7/8


)(2

2

11

t Ot

f f

t

f n

j

n

j!+

!

"=

#

# "+

Leap Frog Method

The simplest stable second-order accurate (in time) method:

Modified equation

( )n j

n

j

n

j

n

j f f h

t U f f

11

11

!+

!+!

"!=

( ) !+!="

"+

"

" xxx

f Uh

x

f U

t

f 1

6

2

2

#

- Stable for 1

- Dispersive (no dissipation) – error will not damp out

- Initial conditions at two t ime levels- Oscillatory solution in time (alternating)

! =U "t

h



!+!

"

"+

!

"

"+!

"

"+=!+

62)()(

3

3

32

2

2 t

t

f t

t

f t

t

f t f t t f

Lax-Wendroff’s Method (LW-I)

First expand the solution in time

Then use the original equation to rewrite the time derivatives

x

f U

t

f

!

!"=

!

!

2

2

2

2

2

x

f U

t

f

xU

x

f U

t t

f

t t

f

!!

=

!!

!!

"=# $ %

&' (

!!

!!

"=# $ %

&' ( !!

!!

=

!!



)(2

)()( 32

2

22 t O

t

x

f U t

x

f U t f t t f !+

!

"

"+!

"

"#=!+

Substituting

Using central differences for the spatial derivatives

f j

n+1= f j

n!U "t

2h f

j +1

n! f j !1

n( ) +U 2"t 2

2h2

f j +1

n! 2 f j

n+ f j !1

n( )

2nd order accurate in space and time

Stable for 1<!

h

t U



Two-Step Lax-Wendroff’s Method (LW-II)

LW-I into two steps:

For the linear equations, LW-II is identical to LW-I

02/

2/)(11

2/1

2/1=

!+

"

+!++

+

+

h

f f U

t

f f f n

j

n

j

n

j

n

j

n

j

0

2/1

2/1

2/1

2/1

1

=

!+

"

! +

!

+

+

+

h

f f U

t

f f n

j

n

j

n

j

n

j

Step 1 (Lax)

Step 2 (Leapfrog)

- Stable for 1/

- Second order accurate in time and space



MacCormack Method

Similar to LW-II, without

( )n j

n

j

n

j

t

j f f h

t U f f !

"!=

+1

( )!"#$%

& '('+= '+ t

j

t

j

t

j

n

j

n

j f f

ht U f f f

1

1

2

1

Predictor

Corrector

- A fractional step method - Predictor: forward differencing - Corrector: backward differencing

- For linear problems, accuracy and stability

properties are identical to LW-I.

2/1,2/1 !+ j j



Second-Order Upwind Method

Warming and Beam (1975) – Upwind for both steps

( )n j

n

j

n

j

t

j f f h

t U f f

1!!

"!=

f j

n+1=

1

2 f

j

n+ f

j

t !U "t h

f j

t ! f j!1

t ( ) !U "t h

f j

n ! 2 f j!1

n+ f

j!2

n( )#$% &

'(

Predictor Corrector

Combining the two:

f j

n+1= f

j

n! " f

j

n! f

j!1

n( ) +1

2" (" ! 1) f

j

n! 2 f

j!1

n+ f

j!2

n( )

- Stable if - Second-order accurate in time and space

20 !! "



8/8


The one-step Lax-Wendroff is not easily extended tonon-linear or multi-dimensional problems. The splitversion is.

In the Lax-Wendroff and the MacCormack methodsthe spatial and the temporal discretization are notindependent.

Other methods have been developed where the time

integration is independent of the spatial discretization,

such as the Beam-Warming and various Runge-Kuttamethods



FTCSUnconditionally

Unstable

UpwindStable for

ImplicitUnconditionally

Stable

Lax-Friedrichs Conditionallyconsistent

Stable for

0=+ xt

Uf f

02

11

1

=

!+

"

!!+

+

h

f f U

t

f f n jn

j

n

j

n

j ( ) xxx xx f Uh

f U

t 222

2162

! +"#"

01

1

=

!+

"

!!

+

h

f f U

t

f f n jn

j

n

j

n

j

( )

( ) xxx

xx

f Uh

f Uh

1326

12

2

2

+!!

!

" "

"

1!"

( ) xxx xx f Uh

f Uh

2

2

13

1

2! !

! "+#

$ %

&' ( "

( )0

2

1

1

1

1

1

=!

+"

! +

!

+

+

+

h

f f U

t

f f n j

n

j

n

j

n

j xxx xx f t U Uh f

t U !"

#$%

&'+(

' 232

2

3

1

6

1

2

( )0

2

2/

11

11

1

=!

+

"

+!

!+

!+

+

h

f f U

t

f f f

n

j

n

j

n

j

n

j

n

j

1!"

Summary


Leap FrogStable for

Lax-Wendroff IStable for

Lax-Wendroff II

Same as LW-IStable for

MacCormack

Same as LW-IStable for

0=+ xt

Uf f

022

11

11

=

!+

"

!!+

!+

h

f f U

t

f f n j

n

j

n

j

n

j ( ) xxx f Uh

16

2

2

!"

( )

( ) xxxx

xxx

f Uh

f Uh

2

3

2

2

18

16

! !

!

""

""

1!"

1!"

1!"

( )0

2

2

2

2

1122

11

1

=+!

"!

!+

"

!

!+

!+

+

h

f f f t U

h

f f U

t

f f

n

j

n

j

n

j

n

j

n

j

n

j

n

j

02/

2/)(11

2/1

2/1=

!+

"

+!++

+

+

h

f f U

t

f f f n j

n

j

n

j

n

j

n

j

0

2/1

2/1

2/1

2/1

1

=

!+

"

! +

!

+

+

+

h

f f U

t

f f n j

n

j

n

j

n

j 1!"

( )0

1

=!

+"

!+

h

f f U

t

f f n j

n

j

n

j

t

j

0

2/1

1

=

!

+"

+!!

+

h

f f

U t

f f f t j

t

j

t

j

n

j

n

j

Summary


Beam-WarmingStable for

QUICKStable for

ENO

WENO

0=+ xt

Uf f

( ) 022

2

43

212

2

21

1

=+!"

!

+!+

"

!

!!

!!

+

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

f f f h

t U

h

f f f U

t

f f

20 !! "

( )( )

( ) ( ) xxxx

xxx

f Uh

f Uh

! !

! !

"""

""

218

216

2

3

2

f j n+1! f j

n

"t +

U 3 f j

n+ 6 f j !1

n! f j !2

n( ) ! 3 f j +1n+ 6 f j

n! f j !1

n( )8h

1!"

A large number of (conditionally) stable andaccurate methods exists for hyperbolic equationswith smooth solutions

And Many More!

Summary

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