18.336 spring 2009
lecture 19
04/10/09
Finite Volume Methods (FVM)
FD: Ujn ≈ function value u(jΔx, nΔt)
1
�
1
(j+ 2 )Δx
n
FV: Uj ≈ cell average
u(x, nΔt)dx
Δx (j− 12 )Δx
Fluxes through cell boundaries
n
n
Ujn+1 − Ujn Fj+ 12 − Fj− 12
+
=0
Δt
Δx
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Godunov Method
REA = Reconstruct-Evolve-Average
Burgers’ equation
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CFL Condition: Δt ≤ C · Δx
Local RP do not interact
1
If f �� (u) > 0 (convex flux function)
⎧
n
Ujn−1 > us , s > 0
⎪
⎨ f (Uj −1 )
n
f (Ujn )
if Ujn < us , s < 0
Fj−
1 =
⎪
2
⎩
f (Us )
Ujn−1 < us < Ujn
s=
f (unj ) − f (unj −1 )
unj − unj−1
⎫
⎪
⎬
(1)
(2)
⎪
⎭
(3)
Shock speed
f � (us ) = 0
Sonic point
[Burgers’: us = 0 ]
Transsonic
rarefaction
If no transsonics occur, we recover exactly upwind.
⇒ FV
Close Relation
←→
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FD.
High Order Methods
Linear case: Taylor series approach → LW
FD: Larger stencils
FV: Reconstruct with linear, quadratic, etc. functions in each cell.
σj n = 0
⇒ Godunov’s method
unj +1 − unj
Δx
⇒ Lax-Wendroff
σj n =
�
��
�
ũn (x,tn )=Ujn +σin ·(x−xj )
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2
Riemann Problem
High order not TVD ⇒ need limiters.
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Nonlinear Stability
Property
Monotone
⇓
L -contracting
⇓
1
Conservation law
Initial conditions
v0 (x) ≥ u0 (x) ∀x
⇒ v(x, t) ≥ u(x, t) ∀x, t.
||u(·, t2 )||L1 ≤ ||u(·, t1 )||L1
∀t2 ≥ t1
Numerical scheme
Vj n ≥ Ujn ∀j
⇒ Vj n+1 ≥ Ujn+1 ∀j
||U n+1 − V n+1 ||1 ≤�
||U n − V n ||1
[||U ||1 = Δx
|uj |]
j
TVD
⇓
TV(u(·, t2 )) ≤ TV(u(·, t1 ))
∀t2 �≥ t1
[TV(u) =
|u(x)|dx]
TV(U
n+1
[TV(U ) =
) ≤ TV(U n )
�
|Uj+1 − Uj |]
j
Monotonicity
preserving
TVB
(bounded)
ux (·, t1 ) ≥ 0 ⇒ ux (·, t2 ) ≥ 0
if t2 > t1
n
Ujn ≥ Uj+1
∀j
n+1
n+1
⇒ Uj ≥ Uj+1 ∀j
TV(U n+1 ) ≤ (1+αΔt)· TV (U n )
α independent of Δt
Remark: Discontinuous solution ⇒ L1 −norm is appropriate
||uh (x) − u(x)||L∞ = 1 ∀h
h→0
||uh (x) − u(x)||L1 −→ 0
Theorem (Godunov):
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A linear, monoticity preserving method is at most first order accurate.
⇒ Need nonlinear schemes.
3
High Resolution Methods
A. Flux Limiters
ut + (f (u))x = 0
Ujn+1 − Ujn Fjn − Fjn−1
+
=0
Δt
Δx
Use two fluxes:
• TVD-flux (e.g. upwind) F̂
• High order flux F̃
Smoothless indicator:
�
uj − uj−1
≈1
θj =
away from 1
uj+1 − uj
where smooth
near shocks
Flux: Fj = F̂j + (F̃j − F̂j ) · Φ(θj )
Ex.: ut + cux = 0
�
�
Δt
Fj = cUj + 2c 1 − c Δx
· (Uj+1 − Uj ) · Φ(θj )
���� �
��
�
Fupwind
FLW −Fupwind
Conditions for Φ(θ):
• TVD: 0 ≤ Φ(θ) ≤ 2θ
0 ≤ Φ(θ) ≤ 2
• Second order: Φ(1) = 1
Φ continuous
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Two Popular Limiters:
Superbee
van Leer
Φ(θ) = max(0, min(1, 2θ), min(θ, 2))
4
Φ(θ) =
|θ| + θ
1 + |θ|
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B. Slope Limiters
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Upwind: σj = 0
Uj+1 − Uj
LW: σj =
Δx
Minmod-limiter:
�
�
Uj − Uj−1 Uj+1 − Uj
,
σj = minmod
Δx
Δx
⎧
⎫
|a| < |b| & ab > 0
⎬
⎨
a
minmod(a, b)
=
b if |a| > |b| & ab > 0
⎩
⎭
0
ab < 0
Many more...
Slope limiters
relation
←→
Flux limiters
5
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18.336 Numerical Methods for Partial Differential Equations
Spring 2009
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