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)