A Second Step Toward FVMs For Systems Of Equations

EP711 Supplementary MaterialThursday, March 8, 2012

Jonathan B. Snively Embry-Riddle Aeronautical University

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Contents

• Linear Solutions• Implementation

EP711 Supplementary MaterialThursday, March 8, 2012

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Example: AcousticsThe linear acoustics equations are given in nonconservative form as:

@p

@t

+ �p0@u

@x

= 0

@u

@t

+1

⇢0

@p

@x

= 0

The system can also be expressed as:

@

@t

p

u

�+

0 �p0

1/⇢0 0

�@

@x

p

u

�= 0

A =

0 �p0

1/⇢0 0

�Thus:

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Acoustics Characteristics

The eigenvalues of the matrix A =

0 �p0

1/⇢0 0

�

�1 = �p

�p0/⇢0 = �c0

�2 = +p

�p0/⇢0 = +c0(Speed of Sound)

The eigenvectors (assembled into a matrix) are given by:

R =�⇢0c0 ⇢0c0

1 1

�

which relate the amplitudes of the characteristic waves to physical quantities in the solution state vector.

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Riemann ProblemsThe Riemann problem is an initial value problem given by the discontinuity at x=0, equal to ql for x<0 and qr for x>0.

The solution is decomposed into characteristics:

qr � ql =mX

p=1

(wpr � wp

l )rp =mX

p=1

↵prp.

For two adjacent cells within a finite volume method solution, this is given by:

Qi �Qi�1 =mX

p=1

↵pi�1/2r

p,

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We aim to find coefficients ...

Which satisfy:

↵p

For a linear system, this is equivalent to (qr � ql) = R↵

↵ = R�1(qr � ql)So, we can obtain coefficients via:

R =�⇢0c0 ⇢0c0

1 1

�R�1 =

12⇢0c0

�1 ⇢0c0

1 ⇢0c0

�

qr � ql =mX

p=1

(wpr � wp

l )rp =mX

p=1

↵prp.

↵ =1

2⇢0c0

�1 ⇢0c0

1 ⇢0c0

� pr � pl

ur � ul

�=

"�(pr�pl)+⇢0c0(ur�ul)

2⇢0c0(pr�pl)+⇢0c0(ur�ul)

2⇢0c0

#

↵1 =�(pr � pl) + ⇢0c0(ur � ul)

2⇢0c0↵2 =

(pr � pl) + ⇢0c0(ur � ul)2⇢0c0

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Contents

• Linear Systems• Implementation

EP711 Supplementary MaterialThursday, March 8, 2012

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LeVeque’s Method

Q

n+1i = Q

ni �

�t

�x

(A+�Qi�1/2 +A��Qi+1/2)��t

�x

(F̃i+1/2 � F̃i�1/2)

Left and Right Fluctuations(Godunov’s Method)

Higher-Order Terms(Flux-Limited via “Waves”)

W̃pi�1/2 = �(✓p

i )Wpi�1/2

Flux-Limited“Wave”

Wave speed(Eigenvalue)

F̃i�1/2 =12

mX

p=1

|spi�1/2|

✓1� �t

�x

|spi�1/2|

◆W̃p

i�1/2High-order

Fluxes:

The “waves” are limited independently – see LeVeque [1997].

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Riemann Problems in LeVeque’s Method and CLAWPACKUsing LeVeque’s notation, define “waves” as

The Riemann solution is then given by:

Wp = ↵prp

Qi �Qi�1 =mX

p=1

Wpi�1/2.

Second: Calculate left and right-going fluctuations given by A+∆q and A-∆q. Here, (s)+ = max(s,0), (s)- = min(s,0).

A±�q =X

p

(�pi�1/2)

±Wpi�1/2

First: Calculate the characteristic coefficients ( ), resulting “waves” ( ), and their “speeds” ( or ).

↵p

Wp �p sp

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LeVeque’s Method and CLAWPACK

Let’s take a look at some examples...

Acoustics Equations (in MATLAB, not CLAWPACK)

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