Finite differences 1

Finite differences

Finite differences 2

Introduction1 j-1 j j+12 N N+1

<------------------------------- L ---------------------------------->

Taylor series expansion:

2 31

2 31

1 1' '' ( ) ''' ( ) ....

2! 3!1 1

' '' ( ) ''' ( ) ....2! 3!

j j j j j

j j j j j

x x x

x x x

1,........, 1j jx x j N

x

Finite differences 3

Finite differences approximations

11 1

1 1 2'' ''' ( ) ..2! 3!

' .j jj j jwhere E x xE

x

forward approximation

=Consistent if , ,…. are bounded

12 2

1 1 2'' ''' ( ) ..2! 3!

' .j jj j jwhere E x xE

x

backward approximationadding both

1 1 1 2''' ( ) ..3!

' .2

j jj jwhere E xE

x

==

=

centered differences

"j '''

j

Finite differences 4

Finite differences approximations (2)

Also

1 1 2 2 44 1' ( )

3 2 3 4j j j j

j O xx x

===

1 1 22

2'' ( )

( )j j j

j O xx

===

fourth order approximation to the first derivative

Second order approximation to the second derivative

Finite differences 5

The linear advection equation00

xU

t00

xU

t

+ initial and boundary conditionsAnalytical solution

)()(),( tTxXtx Substituting we get 0

0 1UC

dt

dT

Tdx

dX

X

U

TUdt

dT

Xdx

dX

0

Eigenvalue problems for the operators

tU

x

eTT

eXX0

0

0

dt

dand

dx

d

With periodic B.C. λ can only have certain (imaginary) valueswhere k is the wavenumber

ik

The general solution is a linear combination of several wavenumbers

Finite differences 6

The linear advection equation (2)

The solution is then:

)(),( 0)(

0)(

0000 tUxfeeTXtx tUxtUx

Propagatingwith speed U0

For a single wave of wavenumber k, the frequency is ω=kU0

No dispersion

Energy conservation

L

dxtE0

2

2

1)(

022 0

20

0

20

L

L Udx

x

U

t

E

If periodic B.C.

Finite differences 7

Space discretization

xxjj

j

211

dt

d

xU

tjjjj

211

0

xikjettTry j )()( ; substituting

0)sin(

0

xk

xkikU

dt

d

whose solution is ikcte 0 with )()sin(

0 kfxk

xkUc

U0

c

kΔx

The phase speed c depends on k; dispersion

kΔx= π ---> λ=2Δx ==> c=0

centered second-order approximation

Finite differences 8

Group velocity

dk

dcg

)cos()(

)(

0*

00

xkUdk

kcdc

Udk

kUdc

g

gContinuous equation

Discretized equation

=-U0 for kΔx=π

Approximating the space operator introduces dispersion

Finite differences 9

Time discretization

tn

t

nj

njj

1

xtU

xU

t

nj

njn

jn

j

nj

nj

nj

nj

2211

0111

0

1

Try xikjetnin

j e 0 Substituting we get

)sin(0 xkix

tUtie

α (Courant-Friedrich-Levy number)

\--v--/

ω=a+ib

If b>0, φjn increases exponentially with time (unstable)

If b<0, φjn decreases exponentially with time (damped)

If b=0, φjn maintains its amplitude with time (neutral)

Also another dispersion is introduced, as we have approximated the operator ∂/ ∂t

first-order forward approx.

Finite differences 10

Three time level scheme (leapfrog))( 11

11

jn

jnn

jn

j)( 11

11

jn

jnn

jn

j

This scheme is centered (second order accurate) in both space and time

Try a solution of the form

xikjnk

nj e 0

exponentialIf |λk| > 1 solution unstableif |λk| = 1 solution neutralif |λk| < 1 solution damped

Substituting

)sin(0122 xkpwhereip kk

21 pipk 11

11

2

2

pip

pip

k

k Δx--->0Δt --->0

physical mode

computational mode

Finite differences 11

Stability analysisEnergy method

define ;2

1)(

0

2L

dxtE

L

LUdx

x

U

t

E

00

202

0 0]22Periodic boundaryconditionsDiscretized analog : En

N

j

nj

n xE1

2)(2

1 φn N+1≡ φn

1

If En=const, stable t

Finite differences 12

Example of the energy method

xU

t

jnn

jn

jn

j

10

1upwind if U0>0downwind if U0<0

xj-1 j

x

tUj

nnj

nj

nj

nj

nj

01

1221 ;))(()()(

21

21

2221 ))(1(})(){()()( j

nnj

nj

nj

nj

nj

j

0

1

21

1 )()1(

N

jj

nj

nnn xEE

En+1=En ifα=0 ==> U0=0 no motionα=1 Δt= Δx/U0

if En+1 > En -------> unstable

En+1 < Enα > 0 ==> U0 > 0 (upwind)α < 1 U0 Δt/ Δx < 1 (CFL condition) damped

Finite differences 13

Von Neumann methodConsider a single wave jikxn

kknj ectx ),(

if |λk| < 1 the scheme is damping for this wavenumber kif |λk| = 1 k the scheme is neutralif |λk| > 1 for some value of k, the scheme is unstable

alternatively jikxtniknj eectx ),(

if Im(ω) > 0 scheme unstableif Im(ω) = 0 scheme neutralif Im(ω) < 0 scheme damping

Vf= ω/k vg=∂ω/∂k

Finite differences 14

Matrix methodLet

nn A 1for a two-time-level scheme

A is called the amplification matrix

kVAnd call the eigenvectors of kkk VVAA

Expanding the initial condition 0 In terms of these eigenvectors

k

kkV00and applying n times the amplification matrix

k

kn

kkn V0

exponential

therefore 1 kn anyif

Finite differences 15

Stability of some schemes• Forward in time, centered in space (FTCS) scheme

• Upwind or downwind

xU

t

nj

nj

nj

nj

211

0

1

using Von Neumann, we find

kxki kk 1)sin(1 scheme unstable

xU

t

nj

nj

nj

nj

10

1upwind if U0 > 0downwind if U0 < 0

Using Von Neumann:

))cos(1()1(21;)1(1

0

2

xke kxik

k

α(α-1) > 0 ------> unstableα < 0 downwindα > 1 CFL limit

-1/4 < α(α-1) < 0==> 0 ≤ α ≤ 1 -------------> stable damped scheme

Finite differences 16

Stability of some schemes (cont)

• Leapfrog

xU

t

nj

nj

nj

nj

2211

0

11

Using von Neumann we find |α|≤1 as stability condition

Finite differences 17

• Lax Wendroff

Stability of some schemes (cont)

From 2

22)(

!2

1),(),(

tt

tttxttx

(Taylor in t)

2

220

20 )(

!2

1),(),(

xUt

xtUtxttx

)2(2

)(2 11

2

111 n

jnj

nj

nj

nj

nj

nj

)(

)(2/)(2/1

2/12/1

2/12/1

2/12/1

1

11

nj

nj

nj

nj

nj

nj

nj

nj

nj

Equivalent to

Applying Von Neumann we can find that |α| ≤1 -----> stable

j j+1/2 j+1

x

Finite differences 18

Stability of some schemes (cont)• Implicit centered scheme

• Krank-Nicholson

xU

t

tj

tj

nj

nj

22

21

21

0

11where

2

112

nnt

using von Neumann

1)sin(1

)sin(12

xki

xki Always neutral

)tan(0 t

t

U

c

Dispersion worsethan leapfrog

xU

t

tj

tj

nj

nj

2

2/12/1

110

1

2

12/1

nnt

where

1)sin(

21

)sin(2

12

xki

xki Always neutral

)2/tan(

2/

0 t

t

U

c

Dispersionbetter thanimplicit.No computationalmode

Finite differences 19

“Intuitive” look at stabilityIf the information for the future time step “comes from” insidethe interval used for the computation of the space derivative,the scheme is stableOtherwise it is unstable

x--> point where the informationcomes from (xj-U0Δt)

j-1 j j+1

x

U0Δt

Interval used for thecomputation of ∂φ/∂x

Downwind scheme

j-1 j j+1

xox ----> α < 1o ----> α > 1

Upwind scheme

CFL number ==> fractionof Δx traveled in Δt secondsLeapfrog

xo

Implicitj-1 j j+1

Finite differences 20

Dispersion and group velocity

ωΔt

π/2 π

U0

vg

vf

Leapfrog

K-N

Implicit

Finite differences 21

Effect of dispersionIn

itial

Lea

pfro

gim

plic

it

Finite differences 22

Two-dimensional advection equation000

yV

xU

t

Using von Neumann, assuming a solution of the form )(0 lykxinn e

we obtain

y

ylV

x

xkUt

)sin()sin( 00

using )sin,cos(),( 00 RRVUV

we obtain, for |λ| to be ≤1 the condition

2R

st

where Δs= Δx= Δy

This is more restrictive than in one dimension by a factor 2