discrete variational derivative methods: geometric integration methods for pdes chris budd (bath),...
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Discrete variational derivative methods:
Geometric Integration methods for PDEs
Chris Budd (Bath), Takaharu Yaguchi (Tokyo),Daisuke Furihata (Osaka)
Have a PDE with solution u(x,y,t)
Variational structure (Lagrangian)*
Conservation laws *
Symmetries linking space and time
Maximum principles
)...,,,,,( yyxxyxt uuuuuFu =
Seek to derive numerical methods which respect/inherit qualitative features of the PDE including localised pattern formation
Cannot usually preserve all of the structure and
Have to make choicesNot always clear what the choices should be
BUT
GI methods can exploit underlying mathematical links between different structures:
Well developed theory for ODEs, supported by backward error analysis
Less well developed for PDEs
Talk will describe the Discrete Variational Derivative Method which works well for PDEs with localised solutions and exploits variational structures
Eg. Computations of localised travelling wave solution of the KdV eqn
Runge-Kutta based method of lines
Discrete variational method
Solution has low truncation error
Solution satisfies a variational principle
1. Hard to develop general structure preserving methods for all PDEs so will look at PDEs with a Variational Structure.
Definition, let u be defined on the interval [a,b]
€
J(u) = G(u,ux ) dxa
b
∫
J(u + δu) − J(u) ≡δG
δu∫ δu dx + Ο δu2
( )
eg.
J(u + δu) − J(u) = (Gu∫ δu + Guxδux ) dx = (Gu∫ −
d
dxGux
)δu dx
δG
δu= Gu −
d
dxGux
€
∂u
∂t=
∂
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟α
δG
δu, J = G(u,ux ) dx∫
PDE has a Variational Form if
Example 1: Heat equation
Example 2: Heat equation (again)
€
ut =uxx
G(u) =u2
2,
δG
δu=u, ut =
∂
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟2δG
δu, α = 2
€
ut =uxx
G(u) = −ux
2
2,
δG
δu=uxx , ut =
δG
δu, α = 0
Example 3: KdV Equation
Example 4: Cahn Hilliard Equation
€
ut = − (6 uux + uxxx )
G(u,ux ) = − u3 +ux
2
2,
ut =∂
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟δG
δu, α =1
€
ut =∂ 2
∂x 2p u + r u3 + q uxx( )
G(u,ux ) = pu2
2+ r
u4
4− q
ux2
2, α = 2
Example 5: Swift-Hohenberg Equation
€
ut = −(1+ ∂x2)2 u + ru + μu2 − γu3
G(u) = −u2
2+ ux
2 −uxx
2
2+ r
u2
2+ μ
u3
3− γ
u4
4
€
ut =δG
δu, α = 0.
Integral of G is the Lagrangian L.
Variational structure is associated with dissipation or conservation laws:
Theorem 1: If
€
α =0,2, −1( )1+α / 2
dJ /dt ≤ 0, α =1 dJ /dt = 0.
Proof:
€
dJ /dt = dG /dt dx∫ = Gu∫ ut + Guxuxt dx =
δG
δu∫ ut dx
=δG
δu∫ ∂
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟α
δG
δudx
α = 0 ⇒ dJ /dt =δG
δu
⎛
⎝ ⎜
⎞
⎠ ⎟2
dx ≥ 0,∫
α =1 ⇒ dJ /dt =1
2
δG
δu
⎛
⎝ ⎜
⎞
⎠ ⎟2 ⎡
⎣ ⎢
⎤
⎦ ⎥ = 0,
α = 2 ⇒ dJ /dt = − ∂x
δG
δu
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝ ⎜
⎞
⎠ ⎟∫2
dx ≤ 0.
€
ut ∂G /∂ux[ ]a
b= 0
2. Discrete Variational Derivative Method (DVDM)
[B,Furihata,Ide,Matsuo,Yaguchi]
Aims to reproduce this structure for a discrete system.
1.Describe method
2. Give examples including the nonlinear heat equation
3. (Backward) Error Analysis
Idea:
Discrete ‘energy’
Discrete integral and discrete integration by parts €
U nk ≈ u(nΔt,kΔx)
€
Gd (Ukn ) ≈ G(u(nΔt,kΔx))
€
Jd (U) − Jd (V ) = TδGd
δ(Uk,Vk )∑ (Uk −Vk ) Δx
Define:
Where the integral is replaced by the trapezium rule
€
J nd = T Gd U n
k( )k
∑ Δx
Now define the Discrete Variational Derivative by:
Discrete Variational Derivative Method
€
Ukn +1 −Uk
n
Δt=
δGd
δ Ukn +1,Uk
n( )€
α =0
Some useful results
Definitions
€
δ+ Uk =Uk +1 −Uk
Δx, δ− Uk =
Uk −Uk−1
Δx, δ (1) Uk =
Uk +1 −Uk−1
2Δx
δ (2) Uk = δ +δ− Uk =Uk +1 − 2Uk + Uk−1
Δx 2
Summation by parts
€
T (δ +∑ fk ) gk = − T fk∑ (δ− gk )
T fk∑ δ (2) gk = − T δ−∑ fk δ−gk
€
Gd = f l∑ (Uk ) gl+(δk
+Uk ) gl−(δk
−Uk ),df
d(U,V )=
f (U) − f (V )
U −V
€
δGd
δ(Uk,Vk )=
df l
d(Uk,Vk )l
∑ gl+(δk
+Uk ) gl−(δk
−Uk ) + gl+(δk
+Vk ) gl−(δk
−Vk )
2
−δk+W l
−(Uk,Vk ) −δk−W l
+(Uk,Vk )
Generally [Furihata], if
€
W l±(Uk,Vk ) =
f l (Uk ) + f l (Vk )
2
⎛
⎝ ⎜
⎞
⎠ ⎟gl
m(δ mUk ) + glm(δ mVk )
2
⎛
⎝ ⎜
⎞
⎠ ⎟
dgl±
d(δk±Uk,δk
±Vk )
Example 1:
€
G(u) = −ux
2
2+ F(u) ≈ Gd (Uk ) = −
1
2
(δ +Uk )2 + (δ−Uk )2
2
⎧ ⎨ ⎩
⎫ ⎬ ⎭+ F(Uk )
€
δGd
δ(Uk,Vk )= δ (2) Uk + Vk
2
⎛
⎝ ⎜
⎞
⎠ ⎟+
dF
d(Uk,Vk )
Heat equation F(u)=0:
€
Ukn +1 −Uk
n
Δt= δ (2) Uk
n +1 + Ukn
2
⎛
⎝ ⎜
⎞
⎠ ⎟
Crank-Nicholson Method
knn
dk
nk
nk
UU
G
t
UU
),( 1)(
1
+
+
=Δ−
δδδ α
€
∂u
∂t=
∂
∂x
⎛
⎝ ⎜
⎞
⎠ ⎟α
δG
δu,
More generally, if
Set
Eg. KdV
€
Ukn +1 −Uk
n
Δt= δk
(1) δGd
δ(Ukn +1,Uk
n )= δk
(1) (Ukn )2 + Uk
nUkn +1 + (Uk
n +1)2 + δk(2) Uk
n + Ukn +1
2
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Jd (U n +1) − Jd (U n ) = TδGd
δ(U n +1,U n )k
∑ (U n +1 −U n )Δx = T δ (α ) δGd
δ(U n +1,U n )k
⎛
⎝ ⎜
⎞
⎠ ⎟
δGd
δ(U n +1,U n )k
∑ ΔxΔt
> 0 α = 0, = 0 α =1, < 0 α = 2
Conservation/Dissipation Property
A key feature of DVDM schemes is that they inherit the conservation/dissipation properties of the PDE and hence have nice stability propertiesTheorem 2: For any N periodic sequence satisfying DVDM
€
Jd (U n +1) − Jd (U n ), > 0 α = 0, = 0 α =1, < 0 α = 2
Proof.
by the summation by parts formulae
Example 2: Nonlinear heat equation
42,0)1()0(,
423 uu
Guuuut
u xxxxx +−===+=
∂∂
€
Ukn +1 −Uk
n
Δtn
=1
2δ (2) Uk
n +1 + Ukn
( ) +1
4Uk
n +1( )
3+ Uk
n +1( )
2Uk
n + Ukn +1 Uk
n( )
2+ Uk
n( )
3
( )
Implementation : Can prove this has a solution if time step small enough: choose this adaptively
• Predict solution at next time step using a standard implicit-explicit method
• Correct using a Powell Hybrid solver
G
U
U
n
J(U)
t
x
u
Example 3: Swift-Hohenberg Equation
€
ut = −(1+ ∂x2)2 u + ru + μu2 − γu3
G(u) = −u2
2+ ux
2 −uxx
2
2+ r
u2
2+ μ
u3
3− γ
u4
4
€
U n +1 −U n
Δt= −
1
21+ δ (2)( )
2U n +1 + U n
( ) + f U n +1,U n( ),
f (u,v) =r
2(u + v) +
μ
3u2 + uv + v 2
( ) −γ
4u3 + u2v + uv 2 + v 3
( ).
€
r = −1, μ =1, γ = −0.15, x ∈ [0,10π ], Periodic BCs
u(x,0) =10 * rand(0,1)
€
L =u2
2∫ − ux
2 +uxx
2
2+ r
u2
2+ μ
u3
3− γ
u4
4dx
3. Backward Error Analysis
This gives some further insight into the solution behaviour
Idea: Set
€
U k
n = u (nΔt,kΔx)
Try to find a suitable function and ‘nice’ operator A so that
€
u t = A ∂xα δG
δu €
G (u )
First consider semi-discrete form then fully discrete
Example of the heat equation: Derived scheme
€
Ukn +1 −Uk
n
Δt= δk
(2) Ukn +1 + Uk
n
2
⎛
⎝ ⎜
⎞
⎠ ⎟
This can be considered to be given by applying the Averaging Vector Field (AVF) method to the ODE system
€
dUk
dt= δk
(2) Uk( )
€
Uk (t) = U (y,kΔx)
U t = U xx +Δx 2
12U xxxx + Ο(Δx 4 )
U t = Uxx +Δx 2
12U txx + Ο(Δx 4 )
U t = AU xx + Ο(Δx 4 ), A = 1−Δx 2
12
∂ 2
∂x 2
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
Backward error approximation
Ill-posed equation this satisfiesEquivalent eqn. to same
order
Well posed backward error eqn which we can improve using Pade
Backward Error Equation has a Variational Structure
€
U t = A∂x2 δG
δU , G =
U 2
2
d
dtG dx =
δG
δU ∫∫ U t dx =
δG
δU ∫ A ∂x
2 δG
δU dx = − A1/ 2∂x
δG
δU
⎛
⎝ ⎜
⎞
⎠ ⎟∫2
dx ≤ 0.
With the dissipation law:
Now apply the AVF method to the modified ODE and apply backward error analysis to this:
Set
€
u (n Δt,k Δx) = Ukn + h.o.t.
And apply the backward error formula for the AVF
€
u t = f (u ), f i = f i +Δt 2
12f j
i fkj f k + Ο(Δt 4 )
To give
€
u t = 1+Δt 2
12A∂xx A∂xx
⎛
⎝ ⎜
⎞
⎠ ⎟A ∂xxu
As the full modified equation satisfied by
€
u (n Δt,kΔx) = Ukn + Ο(Δt 4 + Δx 4 )
This equation has a full variational structure!
Variational structure
€
u t = 1+Δt 2
12A∂xx A∂xx
⎛
⎝ ⎜
⎞
⎠ ⎟A∂xx
δu 2
2
⎛
⎝ ⎜
⎞
⎠ ⎟
δu ,
d
dtu 2∫ dx ≤ 0
Can do very similar analysis for the KdV eqn
€
u t = B∂x +Δt 2
12B∂x
δG
δu B∂x
δG
δu B∂x
⎛
⎝ ⎜
⎞
⎠ ⎟δG
δu , G(u ) = − u 3 +
u x2
2−
Δx 2
6uxx
2
€
B = 1−Δx 2
6∂xx
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
€
d
dtG dx = 0∫Conservation law
The modified eqn also admits discrete soliton solutions which satisfy a modified [Benjamin] variational principle
Eg. Computations of localised travelling wave solution of the KdV eqn
Runge-Kutta based method of lines
Discrete variational method
Solution has low truncation error
Solution satisfies a variational principle
Conclusions
• Discrete Variational Derivative Method gives a systematic way to discretise PDEs in a manner which preserves useful qualitative structures
• Backward Error Analysis helps to determine these structures
• Method can be extended (with effort) to higher dimensions and irregular meshes
• ? Natural way to work with PDEs with a variational structure ?
1. Start with a motivating ODE example which will be useful later.
Hamiltonian system
€
dq /dt = ∂H /∂p, dp /dt = −∂H /∂q
Conservation law:
Separable system (eg. Three body problem)€
H( p,q) = C
€
H( p,q) ≡ T( p) + V (q)
Suppose the ODE has the general form
€
du /dt = f (u)
Set
€
U n ≈ u(n Δt)
Averaged vector field method (AVF) discretises the ODE via:
€
U n +1 −U n
Δt= f (1−ξ ) U n + ξ U n +1
( ) dξ0
1
∫
Properties of the AVF:
1. If f(u) = dF/du then
€
U n +1 −U n
Δt=
F U n +1( ) − F(U n )
U n +1 −U n
2. For the separable Hamiltonian system
€
u = (p,q)
€
Qn +1 − Qn
Δt=
T P n +1( ) − T P n
( )
P n +1 − P n ,P n +1 − P n
Δt= −
V Qn +1( ) −V (Qn )
Qn +1 − Qn
3. Cross-multiply and add to give the conservation law
€
T P n +1( ) + V Qn +1
( ) = T P n( ) + V Qn
( )
Backward error analysis of the AVF method
Set
€
U n = u (n Δt)
To leading order the modified equation satisfied by
For the separable Hamiltonian problem this gives
€
du i /dt = f i(u ) +Δt 2
12f, j
i f,kj f k (u ) + Ο Δt 4
( )
€
dq /dt = 1−Δt 2
12TppVqq
⎛
⎝ ⎜
⎞
⎠ ⎟Tp + Ο(Δt 4 ), d p /dt =− 1−
Δt 2
12TppVqq
⎛
⎝ ⎜
⎞
⎠ ⎟Vq + Ο(Δt 4 )
So, to leading order
€
T( p ) + V (q ) = C + Ο(Δt 4 )
Conservation law plus a phase error of
€
Ο Δt 2( )
€
u (t)