Geometric Numerical Integration

Geometric Numerical Integration

(Ernst Hairer, TU M¨unchen, winter 2009/10)

Development of numerical ordinary differential equations

• Nonstiff differential equations(since about 1850), see [4, 2, 1]

Adams (1855), multistep methods, problem of Bashforth (1883) Runge (1895) and Kutta (1901), one-step methods.

• Stiff differential equations(since about 1950), see [5, 2, 1]

Dahlquist (1963), A-stability of multistep methods Gear (1971), backward differentiation code.

• Geometric numerical integration(since 1986), see [3, 7, 6]

structure-preserving integration of differential equations, Hamiltonian, reversible, divergence-free, Poisson systems.

Provisional contents of the lectures

• Hamiltonian systems

– symplectic transformations, theorem of Poincar´e – generating functions

• Symplectic numerical integrators – St¨ormer–Verlet method

– symplectic Runge–Kutta methods – composition and splitting methods

• Backward error analysis

– long-time energy conservation

– perturbed integrable Hamiltonian systems

• Hamiltonian systems on manifolds

– constrained mechanical systems (DAE’s) – numerical integrator RATTLE

• Differential equations with highly oscillatory solutions – Fermi–Pasta–Ulam type problems

– modulated Fourier expansion

1 Hamiltonian system (double well potential)

Consider a differential equation

¨

q =−∇U(q) which can also be written as

˙ q = p

˙

p = −∇U(q) or q˙ = ∇_pH(p, q)

˙

p = −∇_qH(p, q) with H(p, q) = ¹

2p^Tp+U(q).

We have energy conservation along exact solutions:

H p(t), q(t)

=Const

Numerical experiment with U(q) = (q²−1)², taken from http://sma.epfl.ch/∼vilmart/

11/22/09 2:18 PM A quartic Hamiltonian system (double well)

Page 1 of 2 file:///Users/hairer/cours/geo_int/javaanim/martinet.html

init p(0)=0.0

H?

q(0)=0.01 stepsize h=0.05

Start x 1

speed=

30 ms sleep=

order 1: Symplectic Euler

Explicit Euler

qn+1 = qn+h pn

pn+1 = pn−h∇U(qn) Symplectic Euler

qn+1 = qn+h pn

p_n+1 = p_n−h∇U(q_n+1) or qn+1 = qn+h pn+1

p_n+1 = p_n−h∇U(q_n)

2 N-body problem (planetary motion)

Forp_i, q_i ∈R³we consider the Hamiltonian (kinetic plus potential energy) H(p, q) = 1

2

N

X

i=1

1

m_i p^T_ip_i + X

1≤i<j≤N

U_ij kq_i−q_jk

with the gravitational potential

U_ij(r) =−Gm_im_j r

Illustration: sun – jupiter – saturn (http://sma.epfl.ch/∼vilmart/)

11/22/09 5:46 PM The three body problem Sun-Jupiter-Saturn

Page 1 of 2 file:///Users/hairer/cours/geo_int/javaanim/sunjupitersaturn.html

stepsize h= 3 order 1: Explicit Euler

Start restart speed= x 50

40 ms sleep=

Conserved quantities (first integrals):

• Hamiltonian (total energy) H(p, q)

• linear momentum P(p, q) =

N

X

i=1

p_i

• angular momentum L(p, q) =

N

X

i=1

q_i×p_i

3 N-body problem (molecular dynamics)

N-body problem as before, but with Lennard–Jones potential U_ij(r) = 4ε_ij

σ_ij r

12

−σ_ij r

6

Numerical simulation with initial configuration (left picture) and solution after sufficiently long time (right picture), see http://sma.epfl.ch/∼vilmart/

11/22/09 6:20 PM Molecular dynamics simulation: Lennard-Jones potential

Page 1 of 2 file:///Users/hairer/cours/geo_int/javaanim/lennardjones.html

0.05 stepsize h=

order 2: Stormer-Verlet

Start restart speed= x 1

40 ms sleep=

11/22/09 5:47 PM Molecular dynamics simulation: Lennard-Jones potential

Page 1 of 2 file:///Users/hairer/cours/geo_int/javaanim/lennardjones.html

0.05 stepsize h=

order 2: Stormer-Verlet

Start restart speed= x 1

40 ms sleep=

4 Idea of backward error analysis

For a differential equation y˙ =f(y) consider a numerical solution obtained by a one-step method y_n+1 = Φ_h(y_n).

Find amodified differential equation y˙ =f_h(y) of the form

˙

y=f(y) +hf₂(y) +h²f₃(y) +. . . ,

such that its solution ey(t) satisfies (formally) y_n=y(nh)e .

Numerical experiment with double well potential.

The following pictures show the exact solution (in phase space(q, p)) and solu- tions of the modified equation for various numerical integrators.

exact solution explicit Euler

symplectic Euler symplectic Euler

explicit inp, implicit inq explicit inq, implicit inp Modified differential equation for explicit Euler:

q˙

˙ p

= p

−U⁰(q)

+ h 2

U⁰(q) U⁰⁰(q)p

+ h²

4

−2U⁰⁰(q)p 2U⁰U⁰⁰−U⁰⁰⁰p²

+. . .

Modified differential equation for symplectic Euler (expl. inq, impl. inp):

q˙

˙ p

= p

−U⁰(q)

+ h 2

−U⁰(q) U⁰⁰(q)p

+ h²

12

2U⁰⁰(q)p

−2U⁰U⁰⁰−U⁰⁰⁰p²

+. . .

this modified equation is Hamiltonian with H(p, q) =e 1

2p²+U(q)− h

2U⁰(q)p+h² 12

U⁰(q)²+U⁰⁰(q)p² +. . .

5 Constrained mechanical system

Graphics: J.-P. Eckmann & M. Hairer Position of corners

q_i ∈R³, i= 0, . . . ,5 (6×3 = 18variables) Constraints

• motion of one piece (red) is prescribed, i.e., q0, q1, q2 are fixed (9 conditions),

• distances between neighbor corners is unity (4 conditions),

• orthogonality between neigh- bor edges (qn−1 − q_n) ⊥ (q_n+1−q_n) (5 conditions).

Dynamics(red piece is fixed, corners have equal mass one, massless edges) H(p, q) = 1

2

5

X

i=3

p^T_ip_i+

5

X

i=3

q_iz,

whereq_iz is the vertical component ofq_i.

We obtain a differential-algebraic equation (DAE)

˙

qi = ∇piH(p, q)

˙

p_i = −∇_qiH(p, q)−G^T(q)λ 0 = g(q)

Here, q is the vector composed by q₃, q₄, q₅ , p = ˙q is the velocity, g(q) = 0 represents the 9 constraints forq, andG(q) =g⁰(q)(matrix of dimension9×9).

6 Euler’s equations of motion for a free rigid body

The angular momentum vectory= (y₁, y₂, y₃)^Tsatisfies

˙

y₁ = (I₃⁻¹−I₂⁻¹)y₃y₂

˙

y₂ = (I₁⁻¹−I₃⁻¹)y₁y₃

˙

y₃ = (I₂⁻¹−I₃⁻¹)y₂y₁

or







˙ y₁

˙ y₂

˙ y₃





=







0 −y₃ y₂ y₃ 0 −y₁

−y₂ y₁ 0











 y₁/I₁ y₂/I₂ y₃/I₃







Hamiltonian: H(y) = 1 2

y₁² I₁+y₂²

I₂+y₃² I₃

, Casimir: C(y) = 1 2

y²₁+y₂²+y²₃

1st picture: integration with explicit Euler

2nd picture: trapezoidal rule with projection onto the sphere 3rd picture: implicit midpoint rule

References

[1] J. C. BUTCHER, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons Ltd., Chichester, 2nd ed., 2008.

[2] P. DEUFLHARD AND F. BORNEMANN, Numerische Mathematik 2, de Gruyter Lehrbuch. [de Gruyter Textbook], Walter de Gruyter & Co., Berlin, revised ed., 2008. Gew¨ohnliche Differentialgleichungen. [Ordinary differen- tial equations].

[3] E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integra- tion. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, 2nd ed., 2006.

[4] E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differen- tial Equations I. Nonstiff Problems, Springer Series in Computational Mathe- matics 8, Springer, Berlin, 2nd ed., 1993.

[5] E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II.

Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin, 2nd ed., 1996.

[6] B. LEIMKUHLER AND S. REICH, Simulating Hamiltonian Dynamics, Cam- bridge Monographs on Applied and Computational Mathematics 14, Cam- bridge University Press, Cambridge, 2004.

[7] J. M. SANZ-SERNA AND M. P. CALVO, Numerical Hamiltonian problems, vol. 7 of Applied Mathematics and Mathematical Computation, Chapman &

Hall, London, 1994.