Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Rocky Mountain Mathematics Consortium Summer School
Conservation Laws & Applications
Lecture II: Finite Volume Methods for 1D Systems
James A. Rossmanith
Department of Mathematics
University of Wisconsin – Madison
June 23rd , 2010
J.A. Rossmanith | RMMC 2010
1/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Outline
1 Finite volume schemes
2 Linear systems
3 Exact Riemann solvers
4 Approximate Riemann solvers
J.A. Rossmanith | RMMC 2010
2/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Outline
1 Finite volume schemes
2 Linear systems
3 Exact Riemann solvers
4 Approximate Riemann solvers
J.A. Rossmanith | RMMC 2010
3/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Finite volume schemes
1D schemes
1D finite volume schemes:
Z x 1
i+
1
2
q (tn , x) dx,
Qn
:=
dude
i
∆x x 1
i−
n+ 1
F̄i− 12 :=
2
1
∆t
Z
tn+1
tn
F (q(t, xi− 1 )) dt
2
2
General 1D finite volume scheme
Qn+1
i
=
Qn
i
»
–
∆t
n+ 1
n+ 1
2
2
−
F̄i+ 1 − F̄i− 1
∆x
2
2
n+ 1
Key question: how to compute numerical fluxes F̄i− 12 ?
2
»
– X
N
N
N
X
X
1
∆t
n+ 2
n+ 1
n
2
Qn+1
=
Q
−
F̄
−
F̄
=
Qn
Conservation:
i
i
1
i
N+ 1
∆x
2
2
i=1
i=1
i=1
J.A. Rossmanith | RMMC 2010
4/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Finite volume schemes
1D schemes
1D finite volume schemes:
Z x 1
i+
1
2
q (tn , x) dx,
Qn
:=
dude
i
∆x x 1
i−
n+ 1
F̄i− 12 :=
2
1
∆t
Z
tn+1
tn
F (q(t, xi− 1 )) dt
2
2
General 1D finite volume scheme
Qn+1
i
=
Qn
i
»
–
∆t
n+ 1
n+ 1
2
2
−
F̄i+ 1 − F̄i− 1
∆x
2
2
n+ 1
Key question: how to compute numerical fluxes F̄i− 12 ?
2
»
– X
N
N
N
X
X
1
∆t
n+ 2
n+ 1
n
2
Qn+1
=
Q
−
F̄
−
F̄
=
Qn
Conservation:
i
i
1
i
N+ 1
∆x
2
2
i=1
i=1
i=1
J.A. Rossmanith | RMMC 2010
4/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Wave propagation method
Scalar conservation laws
Definition (LeVeque, 1997)
i ∆t h
i
∆t h −
A ∆Qi+ 1 + A+ ∆Qi− 1 −
F̃i+ 1 − F̃i− 1
2
2
2
2
∆x
∆x
”
“
0
n
Speed: si− 1 = f Q̄i− 1 , where Q̄i− 1 = Ave(Qn
i , Qi−1 )
Qn+1
= Qn
i −
i
2
Wave:
Fluctuations:
Flux correction:
Smoothness:
2
2
A± ∆Qi− 1 := s±
Wi− 1
i− 1
2
2
2
˛
˛
˛s 1 ˛ ∆t !
“
”
˛
˛
i− 2
1˛
F̃i− 1 := si− 1 ˛ 1 −
Wi− 1 φ θi− 1
2
2
2
2
2
∆x
θi− 1 :=
2
Wi− 3
2
Wi− 1
2
Wave limiter:
J.A. Rossmanith | RMMC 2010
2
n
Wi− 1 := Qn
i − Qi−1
φ = 0 (Upwind),
or
Wi+ 1
2
Wi− 1
2
φ = 1 (Lax-Wendroff)
5/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Lax-Wendroff theorem
Theorem (Lax & Wendroff, 1960)
Suppose the method is conservative and consistent with
q,t + f (q),x = 0,
in the sense that
Fi− 1 = F (Qi−1 , Qi )
2
with
F (q̄, q̄) = f (q̄),
and F is Lipschitz continuous.
If a sequence of discrete approximations converge to a function q(t, x) as
∆t, ∆x → 0, then this function is a weak solution of the conservation law.
Notes:
Does not guarantee that there exists a convergent sequence
Two sequences might converge to different weak solutions
To resolve these need: (1) stability and an (2) entropy condition
J.A. Rossmanith | RMMC 2010
6/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Outline
1 Finite volume schemes
2 Linear systems
3 Exact Riemann solvers
4 Approximate Riemann solvers
J.A. Rossmanith | RMMC 2010
7/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Linear hyperbolic systems
Linear system of m hyperbolic PDEs:
q,t + A q,x = 0
System is hyperbolic if A has only real e-vals & is diagonalizable
“
”
A = RΛR−1 , Λ = diag λ(1) , λ(2) , · · · , λ(m)
˛ ˛
˛
˛ ˛
˛
i
i
h
h
˛ ˛
˛
˛ ˛
˛
R = r(1) ˛ r(2) ˛· · · ˛ r(m) , R−T = `(1) ˛ `(2) ˛· · · ˛ `(m)
Diagonalization:
q,t + RΛR−1 q,x = 0,
w := R−1 q
R−1 q,t + ΛR−1 q,x = 0
w,t + Λw,x = 0
=⇒
=⇒
(p)
(p)
w,t + λ(p) w,x
=0
w(p) (t, x) = w̆(p) (x − λ(p) t) = `(p) · q̆(x − λ(p) t)
m n
o
X
∴ q(t, x) =
`(p) · q̆(x − λ(p) t) r(p)
p=1
J.A. Rossmanith | RMMC 2010
8/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Wave propagation framework
Riemann soln. consists of waves W (p) prop. at constant speed λ(p)
Wave decomposition of solution jump
(p)
n
αi− 1 := `(p) · (Qn
i − Qi−1 )
2
Qn
i
−
Qn
i−1
=
m
X
(p)
αi− 1 r(p) :=
2
p=1
Qn+1
= Qn
i −
i
J.A. Rossmanith | RMMC 2010
∆t h
∆x
(3)
m
X
p=1
(3)
(p)
Wi− 1
2
(1)
λ(2) Wi− 1 + λ(2) Wi− 1 + λ(1) Wi+ 1
2
2
i
2
9/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Upwind wave propagation method
First-order wave propagation method
i
∆t h +
Qn+1
A ∆Qi− 1 + A− ∆Qi+ 1
= Qn
i −
i
2
2
∆x
m h
i−
X
(p)
−
(p)
A ∆Qi− 1 =
λ
Wi− 1
2
2
p=1
A+ ∆Qi− 1 =
m h
X
2
λ(p)
i+
(p)
Wi− 1
2
p=1
Conservation
A− ∆Qi− 1 + A+ ∆Qi− 1 =
2
m
X
2
p=1
=
m
X
(p)
λ(p) Wi− 1 =
2
m
X
n
(p)
λ(p) `(p) · (Qn
i − Qi−1 ) r
p=1
n
n
n
n
n
r(p) λ(p) `(p) · (Qn
i − Qi−1 ) = A (Qi − Qi−1 ) = f (Qi ) − f (Qi−1 )
p=1
J.A. Rossmanith | RMMC 2010
10/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Lax-Wendroff method for linear systems
Taylor series approximation
1 2
∆t q,tt + O(∆t3 )
2
1
q(t + ∆, x) = q − ∆t A q,x + ∆t2 A2 q,xx + O(∆t3 )
2
Lax-Wendroff method
i
∆t h
n
n
n
A(Qn
Qn+1
= Qn
i+1 − Qi ) + A(Qi − Qi−1 )
i −
i
2∆x
i
∆2 h 2 n
2
n
n
+
A (Qi+1 − Qn
i ) − A (Qi − Qi−1 )
2
2∆x
q(t + ∆, x) = q + ∆t q,t +
Matrix-vector products
n
A(Qn
i − Qi−1 ) =
m
X
n
(p) (p)
`(p) · (Qn
r
i − Qi−1 ) λ
p=1
n
A2 (Qn
i − Qi−1 ) =
m
X
h
i2
n
(p)
r(p)
`(p) · (Qn
i − Qi−1 ) λ
p=1
J.A. Rossmanith | RMMC 2010
11/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Wave propagation method
Definition (LeVeque, 1997)
Qn+1
= Qn
i −
i
Speed:
i ∆t h
i
∆t h −
A ∆Qi+ 1 + A+ ∆Qi− 1 −
F̃i+ 1 − F̃i− 1
2
2
2
2
∆x
∆x
(p)
si− 1 = λ(p)
2
Wave:
(p)
n
Wi− 1 := `(p) · (Qn
i − Qi−1 )
2
Fluctuations:
±
A ∆Qi− 1 :=
m h
X
2
(p)
si− 1
i±
2
p=1
(p)
Wi− 1
2
0
Flux correction:
F̃i− 1
Smoothness:
(p)
θi− 1
2
2
1
˛
˛
˛ (p) ˛
m
˛
˛
∆t
“
”
˛s
˛
i− 1
1 X˛ (p) ˛ B
C (p)
(p)
2
:=
˛si− 1 ˛ @1 −
A Wi− 1 φ θi− 1
2 p=1
∆x
2
2
2
(p)
:=
2
(p)
2
(p)
Wi− 1 · Wi− 1
2
J.A. Rossmanith | RMMC 2010
(p)
Wi− 3 · Wi− 1
2
(p)
or
(p)
Wi+ 1 · Wi− 1
2
(p)
2
(p)
Wi− 1 · Wi− 1
2
2
12/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Outline
1 Finite volume schemes
2 Linear systems
3 Exact Riemann solvers
4 Approximate Riemann solvers
J.A. Rossmanith | RMMC 2010
13/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Shallow water equations
Basic equations
Shallow water equations:
» –
»
–
»
–
h
hu
0
+
=
hu ,t
hu2 + 21 gh2 ,x
−ghb,x
h(t, x) := Fluid layer thickness
u(t, x) := Depth-averaged fluid layer velocity
b(x) := Bottom topography
J.A. Rossmanith | RMMC 2010
14/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Shallow water equations
Hyperbolic structure
Flux Jacobian:
»
0
A(q) = f (q) =
gh − u2
√
Eigenstructure: c := gh
»
–
»
–
u−c
1
1
Λ=
, R=
,
u+c
u−c u+c
0
1
2u
–
R−1 =
»
1 u+c
2c c − u
−1
1
–
Strictly hyperbolic for all h > 0, weakly hyperbolic if h = 0
Both waves are genuinely nonlinear if h > 0:
Lax entropy condition:
J.A. Rossmanith | RMMC 2010
1-shock if
h? > h` ,
∇q λ(p) · r(p) 6= 0
2-shock if
h? > h r
15/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Shallow water equations
Exact Riemann solution
Find h? such that Φ(h? ) = Φr (h? ) − Φ` (h? ) = 0,
r “
8
”
<u − (h − h ) g 1 + 1
?
`
`
2h
2h
?
`
Φ` (h? ) :=
´
`√
√
:
u` + 2 gh` − gh?
r “
8
”
<u + (h − h ) g 1 + 1
r
?
r
2h
2h
?
r
Φr (h? ) :=
`√
´
√
:
ur − 2 ghr − gh?
Newton iteration:
J.A. Rossmanith | RMMC 2010
hk+1
= hk? −
?
where
if
h? > h `
if
h? ≤ h`
if
h? > hr
if
h? ≤ hr
Φ(hk
?)
Φ0 (hk
?)
16/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
1D nonlinear systems
Godunov’s method
Riemann problem:
q,t + f (q),x = 0,
n
q(t , x) =
( n
Qi−1
x < xi− 1 ,
Qn
i
x > xi− 1 .
2
2
Often can find exact solution
Rankine-Hugoniot conditions
Riemann invariants
entropy conditions
At each interface:
J.A. Rossmanith | RMMC 2010
n+1/2
Fi−1/2 := f (h? , u? )
17/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Outline
1 Finite volume schemes
2 Linear systems
3 Exact Riemann solvers
4 Approximate Riemann solvers
J.A. Rossmanith | RMMC 2010
18/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
A different perspective
BGK relaxation
The problem with exact Riemann solutions:
Generally need to solve nonlinear algebraic equations to find q?
Even though q? is exact, overall solution is not
A small interlude: the Boltzmann equation
1
f,t + v f,x = Q(f, f )
ε
f (t, x, v) := distribution function
Q(f, f ) := collision term,
ε := collision frequency
BGK collision operator [Bhatnagar–Gross–Krook, 1954]:
r
»
–
“
”
m
mv 2
Q(f, f ) = f M − f , f M =
exp −
2πkT
2kT
As ε → 0:
f = f M + εf (1) + O(ε2 )
Relaxation:
J.A. Rossmanith | RMMC 2010
collisions drive f towards f M
19/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Relaxation systems for scalar equations
A simple relaxation system:
» –
»
–» –
»
–
1
0
1
q
q
0
+
=
µ ,t
−α β α + β µ ,x
ε f (q) − µ
Idea: as ε → 0, µ → f
=⇒
q,t + f (q),x = 0
Let µ = f + ε µ(1) + O(ε2 ):
First equation:
Second equation:
Chain rule:
Second equation:
Second equation:
Combine:
2
q,t + f,x = −εµ(1)
,x + O(ε )
f,t − αβ q,x + (α + β) f,x = −µ(1) + O(ε2 )
2
f,t = f,q q,t = −f,q
q,x + O(ε)
˘
¯
(1)
2
− µ = −f,q + (α + β) f,q − αβ q,x + O(ε)
− µ(1) = (f,q − α) (β − f,q ) q,x + O(ε)
h
i
q,t + f,x = ε (f,q − α) (β − f,q ) q,x
+ O(ε2 )
,x
Sub-characteristic condition:
J.A. Rossmanith | RMMC 2010
α ≤ f,q ≤ β
20/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Approximate Riemann solvers
Local Lax-Friedrichs
Operator splitting: » –
»
q
0
Solve RP for
+
µ ,t
−α β
Solve
µ,t =
1
ε
(f − µ)
1
α+β
by setting
–» –
q
=0
µ ,x
µ=f
Local Lax-Friedrichs [Rusanov, 1961]:
Take α = −s and β = s =⇒ α + β = 0 and −αβ = s2 :
–» –
»
–
» –
» –
» – »
Qi − Qi−1
1
1
0 1 q
q
= 0 =⇒
= w1
+w2
+ 2
f (Qi ) − f (Qi−1 )
f ,t s
0 f ,x
−s
s
Finite volume method in fluctuation form
˜
∆t ˆ −
Qn+1
= Qn
A ∆Qi+1/2 + A+ ∆Qi−1/2
i −
i
∆x
1
s
A− ∆Qi−1/2 = −sw1 = (f (Qi ) − f (Qi−1 ) − (Qi − Qi−1 )
2
2
1
s
A+ ∆Qi−1/2 = sw2 = (f (Qi ) − f (Qi−1 ) + (Qi − Qi−1 )
2
2
J.A. Rossmanith | RMMC 2010
21/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Approximate Riemann solvers
Local Lax-Friedrichs for systems
Systems of conservation laws:
»
–
» –
»
–» –
1
0
q
0I
I
q
+
=
µ ,t
−α β I (α + β) I µ ,x
ε f (q) − µ
h
i
h
i
Sub-characteristic condition: α ≤ min λ (f,q ) ≤ max λ (f,q ) ≤ β
˛
˛
LLF: α = −s and β = s, where s = max˛λ(f,q )˛
„»
–«
I
λ
= −s, −s, −s, . . . , −s, s, s, s, . . . , s
s2 I
|
{z
} |
{z
}
m eigenvalues
m eigenvalues
Finite volume method in fluctuation form
˜
∆t ˆ −
A ∆Qi+1/2 + A+ ∆Qi−1/2
Qn+1
= Qn
i −
i
∆x
1
s
A− ∆Qi−1/2 = (f (Qi ) − f (Qi−1 ) − (Qi − Qi−1 )
2
2
1
s
A+ ∆Qi−1/2 = (f (Qi ) − f (Qi−1 ) + (Qi − Qi−1 )
2
2
J.A. Rossmanith | RMMC 2010
22/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Approximate Riemann solvers
Roe solver
Roe approximate Riemann solver [Roe, 1981]:
Relaxation system:
» –
»
q
0I
+
¯2
µ ,t
−(J)
` ´
J¯ = fq Q̄
where
–» –
»
–
1
I
q
0
=
2J¯ µ ,x
ε f (q) − µ
` ´
fq Q̄ (Qr − Q` ) = f (Qr ) − f (Q` )
Eigenvector decomposition
»
–
» (1) –
» (m) –
Qi − Qi−1
r
r
= α(1) (1) (1) + · · · + α(m) (m) (m)
f (Qi ) − f (Qi−1 )
s r
s r
2m equations for only m α’s? Only m lin. ind. eqns due to Roe avg.
Finite volume method in fluctuation form
m h
i± n
o
X
A± ∆Qi−1/2 =
s(p)
`(p) · (Qi − Qi−1 ) r(p)
p=1
J.A. Rossmanith | RMMC 2010
23/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Modified Roe solver
Conservation without Roe average
If Q̄ is not the Roe average, then 2m equations for only m unknowns:
»
–
» (1) –
» (m) –
Qi − Qi−1
r
r
= α(1) (1) (1) + · · · + α(m) (m) (m)
f (Qi ) − f (Qi−1 )
s r
s r
Conservation if we only enforce the second set of m equations:
f (Qi ) − f (Qi−1 ) = β (1) r(1) · · · + β (m) r(m)
β (p) = `(p) · (f (Qi ) − f (Qi−1 ))
Z-waves:
Z (p) = β (p) r(p)
X
1 X
A+ ∆Q =
Z (p) +
2 (p)
(p)
p:s
Conservation:
A+ ∆Q + A− ∆Q =
>0
m
X
p:s
Z (p)
=0
Z (p) = f (Qi ) − f (Qi−1 )
p=1
J.A. Rossmanith | RMMC 2010
24/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Wave propagation method
1D nonlinear systems
Definition (LeVeque, 2002)
i ∆t h
i
∆t h −
A ∆Qi+ 1 + A+ ∆Qi− 1 −
F̃i+ 1 − F̃i− 1
2
2
2
2
∆x
∆x
”
“
(p)
(p)
n
si− 1 = λ
Q̄i− 1 , where Q̄i− 1 = Ave(Qn
i , Qi−1 )
Qn+1
= Qn
i −
i
Speed:
2
2
Wave:
Fluctuations:
(p)
Zi− 1
2
:=
(p)
`i− 1
2
F̃i− 1
Smoothness:
(p)
θi− 1
2
2
(p)
(p)
ri− 1
2
:=
X
(p)
Zi− 1 +
(p)
Zi− 3 · Zi− 1
2
(p)
2
(p)
Zi− 1 · Zi− 1
2
J.A. Rossmanith | RMMC 2010
−
f (Qn
i−1 ))
1 X (p)
Z
2
2 p:s=0 i− 21
2
p:s>0
0
1
˛ (p) ˛
˛s 1 ˛ ∆t
m
“
”
“
”
i− 2
1X
(p)
A Z (p)1 φ θ(p)1
:=
sign si− 1 @1 −
i−
i−
2 p=1
∆x
2
2
2
A± ∆Qi− 1 :=
Correction:
·
2
(f (Qn
i )
2
(p)
or
(p)
Zi+ 1 · Zi− 1
2
(p)
2
(p)
Zi− 1 · Zi− 1
2
2
25/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Wave propagation method
A note about the entropy fix
Need an entropy fix if kth wave is genuinely nonlinear and
λ− := λ(k) (Q` ) < 0
λ+ := λ(k) (Qr ) > 0
In this case, will split kth wave into two waves:
s(k) W (k) = α λ− W (k) + βλ+ W (k)
Conservation dictates that:
α+β =1
Therefore
α=
s − λ+
λ− − λ+
A− ∆Q = A− ∆Q + α λ− W (k)
A+ ∆Q = A+ ∆Q + (1 − α) λ+ W (k)
J.A. Rossmanith | RMMC 2010
26/29
Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Example
1D Euler equations
Euler equations:
2
3
2
3
ρ
ρu
2
4ρu5 + 4 ρu + p 5 = 0
E ,t
u(E + p) ,x
E=
J.A. Rossmanith | RMMC 2010
p
1
+ ρu2
γ−1
2
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Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Example
1D Euler equations
2
3
2
3
ρ
ρu
2
4ρu5 + 4 ρu + p 5 = 0
E ,t
u(E + p) ,x
E=
p
γ−1
+ 12 ρu2
Waves: ¬ Sound, ­ Contact, ® Sound
J.A. Rossmanith | RMMC 2010
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Finite volume schemes
Linear systems
Exact Riemann solvers
Approximate Riemann solvers
Example
1D ideal magnetohydrodynamics
2
∂
∂t
3
2
3
ρ
ρu
`
´
2
1
6ρu7
6
7
kBk
2
6 7 + ∇ · 6ρu `⊗ u + p +
´ I − B ⊗ B7
4E 5
4 u E + p + 1 kBk2 − B (u · B) 5 = 0
2
B
u⊗B−B⊗u
∇·B=0
Waves: ¬ & ³ Fast, ­ & ² Alfvén, ® & ± Slow, ¯ Entropy, ° Div
J.A. Rossmanith | RMMC 2010
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