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)
ut  F (u, ux , u y , uxx , u yy , ...)
Seek to derive numerical methods which respect/inherit
qualitative features of the PDE including localised
pattern formation
Variational structure (Lagrangian)*
Conservation laws *
Symmetries linking space and time
Maximum principles
Cannot usually preserve all of the structure and
Have to make choices
Not 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
Solution has low
truncation error
Discrete variational
method
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) 
b
 G(u,u ) dx
x
a
J(u  u)  J(u) 

G
u dx  u 2 
u
eg.
J(u  u)  J(u) 
G
d
 Gu  Gu
u
dx
x
 (Guu  Gux ux ) dx 
 (Gu 
d
Gux ) u dx
dx
PDE has a Variational Form if



u
 G
  
,
t x  u
J
 G(u,u ) dx
x
Example 1: Heat equation

ut  uxx
u2 G
G(u)  ,
 u,
2
u
  2 G
ut   
,  2
x  u
Example 2: Heat equation (again)

ut  uxx
ux 2 G
G(u)  
,
 uxx ,
2
u
ut 
G
,  0
u
Example 3: KdV Equation
ut   (6 u ux  uxxx )
2
ux
3
G(u,ux )   u 
,
2
  G
ut    ,   1
x  u
Example 4: Cahn Hilliard Equation

2
ut  2 p u  r u 3  q uxx 
x
2
u2
u4
ux
G(u,ux )  p
r
q
,  2
2
4
2
Example 5: Swift-Hohenberg Equation
ut  (1 x2 ) 2 u  ru  u 2  u 3
2
2
3
4
u2
u
u
u
u
G(u)    ux2  xx  r    
2
2
2
3
4

G
ut 
,
u
  0.
Integral of G is the Lagrangian L.
Variational structure is associated with dissipation or
conservation laws:
Theorem 1: If
  0,2,
Proof:
ut G/ux a  0
1 / 2
1

dJ /dt 
b
dJ/dt  0,  1
 dG/dt dx   G u
u
dJ/dt  0.

t  Gux uxt dx

G    G
 
dx
 
u x  u
G 2
  0  dJ /dt     dx  0,
u 
2
1 G  
  1  dJ /dt      0,
2 u  
 G 2
  2  dJ /dt     x   dx  0.
 u 
G
 u
ut dx
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’
U n k  u(nt,kx)
Gd (Ukn )  G(u(nt,kx))
Discrete integral and discrete integration by parts

Define:
J n d  TGd U n k x
k
Where the integral is replaced by the trapezium rule
Nowdefine the Discrete Variational Derivative by:
Jd (U)  Jd (V )  T 
Gd
 (U k ,Vk )
(U k  Vk ) x

Discrete
Variational Derivative Method
Ukn 1 Ukn
Gd

t
Ukn 1,Ukn 

 0
Some useful results
Definitions
U k 1 Uk
U k Uk1
Uk 1 Uk1

(1)
 Uk 
,  Uk 
,  Uk 
x
x
2x
U  2U k  Uk1
(2) Uk     Uk  k 1
x 2

Summation by parts

T (  f k ) gk
  T  f k (  gk )
T  f k  (2) gk
  T   f k   gk
Generally [Furihata], if
Gd   f l (U k ) g ( U k ) g ( U k ),

l

k

l

k
df
f (U)  f (V )

d(U,V )
U V
Gd
dfl
gl (kU k ) gl (kU k )  gl (kVk ) gl (kVk )

 (U k ,Vk )
d(U k ,Vk )
2
l
 kW l (U k ,Vk )  kW l (U k ,Vk )
 f l (Uk )  f l (Vk ) gl ( Uk )  gl ( Vk ) 
dgl 
Wl (Uk ,Vk )  

 



2
2

d(k Uk ,k Vk )


Example 1:
ux 2
1 ( Uk )2  (Uk )2 
G(u)  
 F(u)  Gd (Uk )   
 F(Uk )
2
2 
2

U  Vk 
dF
  (2) k


 2  d(Uk ,Vk )
(Uk ,Vk )
Gd

Heat equation F(u)=0:

 n 1  U n 
Uk n 1 Uk n
(2) Uk
k
  

t
2


Crank-Nicholson Method

More generally, if



u
 G
  
,
t x  u
Set

U kn1  U kn
Gd
  k( )
t
 (U n1 ,U n ) k
Eg. KdV
n 1 
 n

 n
Ukn 1  Ukn
Gd
n
n 1
n 1 2
(2) U k  U k
(1)
(1)
2
 k
 k (Uk )  Uk Uk  (Uk )  k 

n 1
n
t
2
(Uk ,Uk )



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 properties
Theorem 2: For any N periodic sequence satisfying DVDM
Jd (U n 1)  Jd (U n ),  0   0,  0  1,  0   2
Proof.
Jd (U
n 1
)  Jd (U )  T 
n
0
Gd
 (U n 1,U n ) k
  0,
(U
0
n 1
 Gd
 Gd
 U )x  T   n 1 n 
xt
n 1
n

(U
,U
)

(U
,U
)

k 
k
n
( )
  1,  0   2
by the summation by parts formulae
Example 2: Nonlinear heat equation
u
 u xx  u 3 ,
t
u x (0)  u x (1)  0,
u x2 u 4
G 
2
4

Ukn 1 Ukn 1 (2) n 1
1
n
n 1 3
n 1 2 n
n 1
n 2
n 3
  Uk  Uk  Uk   Uk  Uk  Uk Uk   Uk 
tn
2
4

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

U
J(U)
G
U
n
t
u
x
Example 3: Swift-Hohenberg Equation
ut  (1 x2 ) 2 u  ru  u 2  u 3
u2
uxx2
u2
u3
u4
2
G(u)    ux 
 r  
2
2
2
3
4

U n 1  U n
1
(2) 2
  1   U n 1  U n  f U n 1,U n ,
t
2
r
 2
 3 2
2
f (u,v)  (u  v)  u  uv  v  u  u v  uv2  v 3 .
2
3
4
r  1,   1,   0.15, x  [0,10 ], PeriodicBCs
u(x,0)  10* rand(0,1)

L


2
u2
uxx
u2
u3
u4
2
 ux 
 r  
dx
2
2
2
3
4
3. Backward Error Analysis
This gives some further insight into the solution behaviour
Idea: Set
U k  u(nt,kx)
n
Try to find a suitable function G (u )
operator A so that

 G
ut  A x
u
and ‘nice’
First consider
 semi-discrete form then fully discrete
Example of the heat equation: Derived scheme
 n 1  U n 
Uk n 1 Uk n
(2) U k
k
 k 

t
2


This can be considered to be given by applying the Averaging
Vector Field (AVF) method to the ODE system

dUk
(2)
 k Uk 
dt
U k (t)  U (y,kx)
Backward error approximation
x 2 
U t  U xx 
U xxxx  (x 4 )
Ill-posed equation this satisfies
12
x 2
Equivalent eqn. to same order
U t  U xx 
U txx  (x 4 )
12
2
2 1

x 
U t  AU xx  (x 4 ), A  1
Well posed backward error eqn
2 
12

x


which we can improve using Pade
Backward Error Equation has a Variational Structure
G
U t  A x
,
U
2
U2
G
2
With the dissipation law:
d
dt

G
G
dx


 U U t dx 
 1/ 2 G 
G
2 G
A

dx


 U x U
 A x U  dx  0.
2
Now apply the AVF method to the modified ODE
and apply backward error analysis to this:
Set u(n t,k x)  Uk  h.o.t.
n
 And apply the backward error formula for the AVF
ut  f (u ),
To give

t 2 i j k
f  f 
f j f k f  (t 4 )
12
i
i
 t 2

ut  1
A xx A xx A xx u
 12

As the full modified equation satisfied by

u(n t,kx)  Ukn  (t 4  x 4 )
This equation has a full variational structure!
Variational structure
u 2 
 
2
 t

 2 
ut  1
Axx Axx Axx
,
u
 12


d
dt
u
2
dx  0
Can do very similar analysis for the KdV eqn

G
t 2
G
G
ux2 x 2 2
3
ut  Bx 
Bx
Bx
Bx  , G(u )   u 

uxx
12
u
u
2
6

 u
 x 2 1
B  1
xx 
6


Conservation law
d
dt
 G dx  0
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
Solution has low
truncation error
Discrete variational
method
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:
H( p,q)  C
Separable system (eg. Three body problem)

H( p,q)  T( p)  V (q)
Suppose the ODE has the general form
du /dt  f (u)
Set
U n  u(n t)

Averaged vector field method (AVF) discretises
the ODE via:
U n 1  U n

t

1
 f (1  ) U
0
n
  U n 1 d
Properties of the AVF:
1. If f(u) = dF/du then
n 1
n
U n 1  U n F U  F(U )

t
U n 1  U n

2. For the separable Hamiltonian system
n 1
n
Qn 1  Qn T P  T P 

,
t
P n 1  P n

u  ( p,q)
V Qn 1V (Qn )
P n 1  P n

t
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
Un
 u(n t)
To leading order the modified equation satisfied by

2
t
du i /dt  f i (u ) 
f, ij f,kj f k (u )  t 4 
12
u(t)

For the separable Hamiltonian problem this gives

 t 2

 t 2

4
d q /dt  1
TppVqq Tp  (t ), d p /dt  1
TppVqq Vq  (t 4 )
 12

 12

4
So, to leading order T( p)  V (q )  C  (t )

Conservation law plus a phase error of

t 2 