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 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 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 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]

PDE has a Variational Form if Example 1: Heat equation Example 2: Heat equation (again)

Example 3: KdV Equation Example 4: Cahn Hilliard Equation

Example 5: Swift-Hohenberg Equation Integral of G is the Lagrangian L.

Variational structure is associated with dissipation or conservation laws: Theorem 1: If Proof:

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

Define: Where the integral is replaced by the trapezium rule Now define the Discrete Variational Derivative by: Discrete Variational Derivative Method

Some useful results Definitions Summation by parts

Generally [Furihata], if

Example 1: Heat equation F(u)=0: Crank-Nicholson Method

More generally, if Set Eg. KdV

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

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

U n J(U)

t

x u

Example 3: Swift-Hohenberg Equation

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

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

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:

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!

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

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 Conservation law: Separable system (eg. Three body problem)

Suppose the ODE has the general form Set Averaged vector field method (AVF) discretises the ODE via:

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

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