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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