Energy Preserving Numerical Integration Methods

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1 Energy Preserving Numerical Integration Methods Department of Mathematics and Statistics, La Trobe University Supported by Australian Research Council Professorial Fellowship Collaborators: Celledoni (Trondheim) Grimm (Erlangen) McLachlan (Massey) McLaren (La Trobe) O Neale (La Trobe) Owren (Trondheim)

2 Outline of the talk Energy-preserving integrators for ODEs

3 Outline of the talk Energy-preserving integrators for ODEs Energy-preserving integrators for PDEs

4 PART I Energy-preserving integrators for Ordinary Differential Equations

5 Consider a Poisson ODE: dx dt = S(x) H(x). Here S(x) is a skew matrix, and H is the gradient of the Hamiltonian energy function.

6 Consider a Poisson ODE: dx dt = S(x) H(x). Here S(x) is a skew matrix, and H is the gradient of the Hamiltonian energy function. Definition (Skew matrix) S(x) is skew if (a S(x)b) = (b S(x)a) a, b

7 ENERGY-PRESERVING DISCRETE GRADIENT METHOD Definition (Discrete gradient) A discrete gradient H(x n, x n+1 ) is defined by (i) (x n+1 x n ) H(x n, x n+1 ) H(x n+1 ) H(x n ) and (ii) lim xn+1 x n H(x n, x n+1 ) = H(x n )

8 Theorem Let H be a discrete gradient. Then the discretization x n+1 x n t = S(x n ) H(x n, x n+1 ) is energy-preserving.

9 Theorem Let H be a discrete gradient. Then the discretization x n+1 x n t = S(x n ) H(x n, x n+1 ) is energy-preserving. Proof. H(x n+1 ) H(x n ) = H(x n, x n+1 ) (x n+1 x n ) = t H(x n, x n+1 ) S(x n ) H(x n, x n+1 ) = 0

10 TWO EXAMPLES OF DISCRETE GRADIENTS Remember (x n+1 x n ) H(x n, x n+1 ) H(x n+1 ) H(x n ) Example 1. (Itoh-Abe discrete gradient): H := ( H(xn+1,y n) H(x n,y n) x n+1 x n H(x n+1,y n+1 ) H(x n+1,y n) y n+1 y n ). (This can be generalised to any dimension)

11 TWO EXAMPLES OF DISCRETE GRADIENTS Remember (x n+1 x n ) H(x n, x n+1 ) H(x n+1 ) H(x n ) Example 1. (Itoh-Abe discrete gradient): H := ( H(xn+1,y n) H(x n,y n) x n+1 x n H(x n+1,y n+1 ) H(x n+1,y n) y n+1 y n ). (This can be generalised to any dimension) Example 2. ( Average" discrete gradient): H := 1 0 H(ξx n+1 + (1 ξ)x n ) dξ

12 Proof. 1 (x n+1 x n ) H = (x n+1 x n ) H(ξx n+1 + (1 ξ)x n ) dξ 0 1 d = 0 dξ H(ξx n+1 + (1 ξ)x n ) dξ = H(ξx n+1 + (1 ξ)x n ) = H(x n+1 ) H(x n ) ξ=1 ξ=0

13 So x n+1 x n t = S(x) 1 0 H(ξx n+1 + (1 ξ)x n ) dξ is an energy preserving integrator for the ODE dx dt = S(x) H(x)

14 So x n+1 x n t = S(x) 1 0 H(ξx n+1 + (1 ξ)x n ) dξ is an energy preserving integrator for the ODE dx dt = S(x) H(x) From now on we take S(x) = S (constant).

15 So x n+1 x n t = S(x) 1 0 H(ξx n+1 + (1 ξ)x n ) dξ is an energy preserving integrator for the ODE dx dt = S(x) H(x) From now on we take S(x) = S (constant). Then x n+1 x n t x n+1 x n t = S = H(ξx n+1 + (1 ξ)x n ) dξ S H(ξx n+1 + (1 ξ)x n ) dξ

16 So x n+1 x n t = S(x) 1 0 H(ξx n+1 + (1 ξ)x n ) dξ is an energy preserving integrator for the ODE dx dt = S(x) H(x) From now on we take S(x) = S (constant). Then x n+1 x n t x n+1 x n t = S = H(ξx n+1 + (1 ξ)x n ) dξ S H(ξx n+1 + (1 ξ)x n ) dξ But note that S H is the vector field! We get:

17 Theorem (The average vector field method") The numerical integration method x n+1 x n t = 1 0 f(ξx n+1 + (1 ξ)x n ) dξ preserves the energy H(x) exactly for any Hamiltonian ODE with constant symplectic structure, i.e. for dx dt = f(x) with f(x) = S H(x)

18 Example (DOUBLE WELL POTENTIAL) dx dt dy dt = y = 2x(1 x 2 )

19 Example (DOUBLE WELL POTENTIAL) AVF integrator dx dt dy dt = y = 2x(1 x 2 ) ( 1 xn+1 x n t y n+1 y n ) = ( 1 2 (y n+1 + y n ) x n+1 + x n 1 2 (x3 n + x 2 nx n+1 + x n x 2 n+1 + x3 n+1 ) )

20 Energy-preserving integrators for Partial Differential Equations

21 Hamiltonian PDEs Definition The PDE ( u t = f u, u ) x, 2 u x 2,... is Hamiltonian if it is of the form ( f u, u ) x, 2 u x 2,... = Ŝ δh δu δh where Ŝ is a constant skew operator, and δu derivative of H. is the variational

22 Hamiltonian PDEs Definition The PDE ( u t = f u, u ) x, 2 u x 2,... is Hamiltonian if it is of the form ( f u, u ) x, 2 u x 2,... = Ŝ δh δu δh where Ŝ is a constant skew operator, and is the variational δu derivative of H. (Note: f, u, and x can also be taken to be vectors. Although u is usually real, it may also be complex).

23 Hamiltonian PDEs So a Hamiltonian PDE is written u t = Ŝ δh δu

24 Hamiltonian PDEs So a Hamiltonian PDE is written u t = Ŝ δh δu Definition (Variational derivative) Let then δh δu H(u) := H ( u, u ) x, 2 u x 2,... dx H := u ( ) ( ) H + 2 H x u x x 2 u xx

25 Hamiltonian PDEs So a Hamiltonian PDE is written u t = Ŝ δh δu Definition (Variational derivative) Let then δh δu H(u) := H ( u, u ) x, 2 u x 2,... dx H := u ( ) ( ) H + 2 H x u x x 2 u xx Definition (skew operator) Ŝ is skew if a.ŝb = b.ŝa, a, b where a.c := a(x)b(x)dx

26 Example (1. sine-gordon equation (a)) 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ

27 Example (1. sine-gordon equation (a)) Continuous: H = Ŝ = 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ [ 1 2 π2 + 1 ( ) ϕ 2 + α(1 cos(ϕ))] dx 2 x ( )

28 Example (1. sine-gordon equation (a)) Continuous: 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ H = Ŝ = Semi-discrete [ 1 2 π2 + 1 ( ) ϕ 2 + α(1 cos(ϕ))] dx 2 x ( ) ( ϕ x ϕ ) n ϕ n 1 x [ 1 H = x 2 π2 n + n ( 0 id S = id 0 1 2( x) 2 (ϕ n ϕ n 1 ) 2 + α (1 cos(ϕ n )) ), denotes standard gradient ]

29 SG equation (B.D.), Nspace= 101, t =0.01 avf2 Midpoint exact SG equation (B.D.), Nspace= 101, t =0.01 avf2 Midpoint Energy Global Error time time Figure: sine-gordon equation, Backward Differences for spatial discretizations : Energy (left) and global error (right) vs time, for AVF and implicit midpoint integrators.

30 Example (2. sine-gordon equation (b)) 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ

31 Example (2. sine-gordon equation (b)) Continuous: H = Ŝ = 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ [ 1 2 π2 + 1 ( ) ϕ 2 + α(1 cos(ϕ))] dx 2 x ( )

32 Example (2. sine-gordon equation (b)) Continuous: 2 ϕ t 2 = 2 ϕ x 2 + α sin ϕ H = Ŝ = Semi-discrete [ 1 2 π2 + 1 ( ) ϕ 2 + α(1 cos(ϕ))] dx 2 x ( ) ( ϕ x ϕ ) n+1 ϕ n 1 2 x [ 1 H = x 2 π2 n + n ( ) 0 id S =. id 0 1 8( x) 2 (ϕ n ϕ n 1 ) 2 + α (1 cos(ϕ n )) ]

33 SG equation (C.D.), Nspace= 101, t =0.01 avf2 Midpoint exact SG equation (C.D.), Nspace= 101, t =0.01 avf2 Midpoint Energy Global Error time time Figure: sine-gordon equation, Central Differences for spatial discretizations : Energy (left) and global error (right) vs time, for AVF and implicit midpoint integrators.

34 Example (3. Korteweg-deVries equation (a)) u t = 6u u x 3 u x 3

35 Example (3. Korteweg-deVries equation (a)) Continuous: u t = 6u u x 3 u x 3 H = Ŝ = x [ ] 1 2 u2 x u 3 dx

36 Example (3. Korteweg-deVries equation (a)) Continuous: u t = 6u u x 3 u x 3 H = Ŝ = x [ ] 1 2 u2 x u 3 dx Semi-discrete: (u 3 u 3 n) H = x [ ] 1 2( x) 2 (u n u n 1 ) 2 u 3 n n S =... 2 x

37 KdV equation, Nspace= 401, t =0.001 avf2 Midpoint exact Energy time Figure: KdV equation: Exact energy, and energy vs time given by AVF and implicit midpoint methods, using discretization (a).

38 Example (4. Korteweg-deVries equation (b)) u t = 6u u x 3 u x 3

39 Example (4. Korteweg-deVries equation (b)) Continuous: u t = 6u u x 3 u x 3 H = Ŝ = x [ ] 1 2 u2 x u 3 dx

40 Example (4. Korteweg-deVries equation (b)) Continuous: u t = 6u u x 3 u x 3 H = Ŝ = x [ ] 1 2 u2 x u 3 dx Semi-discrete: (u 3 u n u n+1 u n+2 ) (for illustrative purposes only) H = x [ ] 1 2( x) 2 (u n u n 1 ) 2 u n u n+1 u n+2 n S =... 2 x

41 Example (4. Korteweg-deVries equation (b)) Continuous: u t = 6u u x 3 u x 3 H = Ŝ = x [ ] 1 2 u2 x u 3 dx Semi-discrete: (u 3 u n u n+1 u n+2 ) (for illustrative purposes only) H = x [ ] 1 2( x) 2 (u n u n 1 ) 2 u n u n+1 u n+2 n S =... 2 x Note: Large freedom in semi-discretisation H without destroying energy preservation. But S must be skew!

42 KdV equation, Nspace= 401, t =0.001 avf2 Midpoint Energy time Figure: KdV equation: Energy vs time given by AVF and implicit midpoint methods, using discretization (b).

43 Example (5. NLS equation) Continuous: ( ) ( u 0 i t u = i 0 ) ( δh δu δh δu ), (1) where u denotes the complex conjugate of u. [ ] H = u 2 x + γ 2 u 4 dx, (2) ( ) 0 i Ŝ =. i 0 (3)

44 Example (5. NLS equation) Continuous: ( ) ( u 0 i t u = i 0 ) ( δh δu δh δu ), (1) where u denotes the complex conjugate of u. [ ] H = u 2 x + γ 2 u 4 dx, (2) ( ) 0 i Ŝ =. i 0 (3) Semi-discrete: H = j [ 1 ( x) 2 u j+1 u j 2 + γ ] 2 u j 4, (4) ( ) 0 id S = i. (5) id 0

45 16 x NLS equation, Nspace= 201, Δt = NLS equation, Nspace= 201, Δt =0.05 avf2 Midpoint Energy Error avf2 Midpoint Global Error time time Figure: Nonlinear Schrödinger equation: Energy error (left) and global error (right) vs time, for AVF and implicit midpoint integrators.

46 14 x 10 6 NLS equation, Nspace= 201, Δt =0.05 Probability Error avf2 Midpt time Figure: Nonlinear Schrödinger equation: Total probability error vs time, for AVF and implicit midpoint integrators.

47 Example (6. Nonlinear Wave Equation) Continuous: H = Ŝ = ϕ t 2 = ( 2 x + 2 y)ϕ ϕ 3 1 ( [ 1 2 (π2 + ϕ 2 x + ϕ 2 y) + 1 ] 4 ϕ4 dx dy )

48 Example (6. Nonlinear Wave Equation) Continuous: H = Ŝ = ϕ t 2 = ( 2 x + 2 y)ϕ ϕ 3 1 ( [ 1 2 (π2 + ϕ 2 x + ϕ 2 y) + 1 ] 4 ϕ4 dx dy ) Semi-discrete: We discretize the Hamiltonian in space with a tensor product Lagrange quadrature formula based on p + 1 Gauss-Lobatto-Legendre (GLL) quadrature nodes in each space direction:

49 Nonlinear Wave Equation II H = 1 2 p j 1 =0 j 2 =0 where d j1,k = dl k(x) dx p w j1 w j2 (π j 2 1,j ( p ) 2 d j1,kϕ k,j2 k=0 ( p m=0 d j2,mϕ j1,m ) ϕ4 j 1,j 2, and l k (x) is the k-th Lagrange basis x=xj1 function based on the GLL quadrature nodes x 0,..., x p, and with w 0,..., w p the corresponding quadrature weights. )

50 T = 0 T = x 10 3 x T = T = x 10 3 x Figure: Snapshots of the solution of the 2D wave equation at different times. AVF method with step-size t = Space discretization with 6 Gauss Lobatto nodes in each space direction. Numerical solution interpolated on a equidistant grid of 21 nodes in each space direction.

51 0.5 x D Wave equation avf and ode15s Energy error time Figure: The 2D wave equation. MATLAB routine ode15s with absolute and relative tolerance (dashed line), and AVF method with step size t = 10/(2 5 ) (solid line). Energy error versus time. Time interval [0, 10]. Space discretization with 6 Gauss Lobatto nodes in each space direction.

52 Leapfrog

53 Exponential Integrator

54 GENERALIZATIONS AND FURTHER REMARKS Instead of using finite differences for the (semi-)discretization of the spatial derivatives one may use e.g. a spectral discretization.

55 GENERALIZATIONS AND FURTHER REMARKS Instead of using finite differences for the (semi-)discretization of the spatial derivatives one may use e.g. a spectral discretization. The method presented also applies to linear PDEs, e.g. the (Linear) Time-dependent Schrödinger equation, 3D Maxwell s equation.

56 GENERALIZATIONS AND FURTHER REMARKS Instead of using finite differences for the (semi-)discretization of the spatial derivatives one may use e.g. a spectral discretization. The method presented also applies to linear PDEs, e.g. the (Linear) Time-dependent Schrödinger equation, 3D Maxwell s equation. The method presented can be generalised to u t = ˆN δh δu where ˆN is a constant negative (semi) definite operator, and where H H is a Lyapunov function, i.e. t 0. (e.g. Allen-Cahn eq, Cahn-Hilliard eq, Ginzburg-Landau eq.)

57 GENERALIZATIONS AND FURTHER REMARKS The method can also be generalised to preserving any number of integrals (not necessarily energy).

58 GENERALIZATIONS AND FURTHER REMARKS The method can also be generalised to preserving any number of integrals (not necessarily energy). If one replaces the integral in the Average Vector Field by quadrature of a certain order, the resulting method will preserve energy to a certain order.

59 GENERALIZATIONS AND FURTHER REMARKS The method can also be generalised to preserving any number of integrals (not necessarily energy). If one replaces the integral in the Average Vector Field by quadrature of a certain order, the resulting method will preserve energy to a certain order. The Average Vector Field Method is a B-series method.

60 Some references to our work ODEs 1. McLachlan, Quispel & Robidoux, A unified approach to Hamiltonian systems, Poisson systems, gradient systems and systems with Lyapunov functions and/or first integrals, Phys. Rev. Lett. 81(1998) McLachlan, Quispel & Robidoux, Geometric integration using discrete gradients, Phil. Trans. Roy. Soc. A357(1999) Quispel & McLaren, A new class of energy-preserving numerical integration methods, J. Phys A41(2008) (7pp). PDEs 4. Celledoni, Grimm, McLachlan, McLaren, O Neale & Quispel, Preserving energy resp. dissipation in numerical PDEs, using the Average Vector Field method, Submitted for publication. Of course there are also many important publications by Furihata, Matsuo, Yaguchi, and collaborators.

norges teknisk-naturvitenskapelige universitet Preserving energy resp. dissipation in numerical PDEs, using the Average Vector Field method

norges teknisk-naturvitenskapelige universitet Preserving energy resp dissipation in numerical PDEs, using the Average Vector Field method by E Celledoni,V Grimm, RI McLachlan, DI McLaren, DRJ O Neale,