Discrete variational derivative methods:

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Discrete variational derivative
methods Geometric Integration methods for PDEs
Chris Budd (Bath), Takaharu Yaguchi
(Tokyo), Daisuke Furihata (Osaka)
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Have a PDE with solution u(x,y,t)
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
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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
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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
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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
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PDE has a Variational Form if
Example 1 Heat equation
Example 2 Heat equation (again)
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Example 3 KdV Equation
Example 4 Cahn Hilliard Equation
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Example 5 Swift-Hohenberg Equation
Integral of G is the Lagrangian L.
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Variational structure is associated with
dissipation or conservation laws
Theorem 1 If
Proof
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• 2. Discrete Variational Derivative Method (DVDM)
• B,Furihata,Ide,Matsuo,Yaguchi
• Aims to reproduce this structure for a discrete
  system.
• Describe method
• Give examples including the nonlinear heat
  equation
• (Backward) Error Analysis

Idea Discrete energy Discrete integral and
discrete integration by parts
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Define
Where the integral is replaced by the trapezium
rule
Now define the Discrete Variational Derivative by
Discrete Variational Derivative Method
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Some useful results
Definitions
Summation by parts
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Generally Furihata, if
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Example 1
Heat equation F(u)0
Crank-Nicholson Method
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More generally, if
Set
Eg. KdV
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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
Proof.
by the summation by parts formulae
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Example 2 Nonlinear heat
equation

• 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

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J(U)
U
n
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t
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u
x
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Example 3 Swift-Hohenberg Equation
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3. Backward Error Analysis This
gives some further insight into the solution
behaviour Idea Set
Try to find a suitable function and
nice operator A so that
First consider semi-discrete form then fully
discrete
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Example of the heat equation Derived scheme
This can be considered to be given by applying
the Averaging Vector Field (AVF) method to the
ODE system
Backward error approximation
Ill-posed equation this satisfies
Equivalent eqn. to same order
Well posed backward error eqn which we can
improve using Pade
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Backward Error Equation has a Variational
Structure
With the dissipation law
Now apply the AVF method to the modified ODE and
apply backward error analysis to this
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Set
And apply the backward error formula for the AVF
To give
As the full modified equation satisfied by
This equation has a full variational structure!
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Variational structure
Can do very similar analysis for the KdV eqn
Conservation law
The modified eqn also admits discrete soliton
solutions which satisfy a modified Benjamin
variational principle
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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
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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 ?

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1. Start with a motivating ODE example which will
be useful later. Hamiltonian system
Conservation law Separable system (eg. Three
body problem)
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Suppose the ODE has the general form
Set
Averaged vector field method (AVF) discretises
the ODE via
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Properties of the AVF 1. If f(u) dF/du then
2. For the separable Hamiltonian system
3. Cross-multiply and add to give the
conservation law
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Backward error analysis of the AVF method
Set
To leading order the modified equation satisfied
by
For the separable Hamiltonian problem this gives
So, to leading order
Conservation law plus a phase error of