Geometric Numerical Integration of Differential Equations Ernst Hairer Universit´ e de Gen` eve Aveiro, September 10 -14, 2018 E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration, Second edition, Springer-Verlag, 2006 Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 1 / 43
Summerschool in Aveiro (Sept. 2018), Ernst Hairer Part I. Geometric numerical integration ◮ Hamiltonian systems, symplectic mappings, geometric integrators, St¨ ormer–Verlet, composition and splitting, variational integrator ◮ Backward error analysis, modified Hamiltonian, long-time energy conservation, application to charged particle dynamics Part II. Differential equations with multiple time-scales ◮ Highly oscillatory problems, Fermi–Pasta–Ulam-type problems, trigonometric integrators, adiabatic invariants ◮ Modulated Fourier expansion, near-preservation of energy and of adiabatic invariants, application to wave equations Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 2 / 43
Lecture 1. Introduction to geometric integration Introduction and examples 1 Explicit, implicit, and symplectic Euler Kepler and N-body problems Hamiltonian systems 2 Symplectic mappings Theorem of Poincar´ e Symplectic Euler methods St¨ ormer–Verlet integrator Symplectic integration methods 3 Runge–Kutta methods Composition and splitting methods Variational integrators Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 3 / 43
Mathematical pendulum q q ˙ = p or q + sin q = 0 ¨ cos q p ˙ = − sin q The energy 2 2 p 2 − cos q H ( p , q ) = 1 1 is constant along solutions: −3 −2 −1 0 1 2 3 4 −1 � � H p ( t ) , q ( t ) = Const. −2 Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 4 / 43
Mathematical pendulum q q ˙ = p or q + sin q = 0 ¨ cos q p ˙ = − sin q Proof. d � � d t H p ( t ) , q ( t ) = p ( t ) ˙ p ( t ) + sin( q ( t )) ˙ q ( t ) = 0 The energy 2 2 p 2 − cos q H ( p , q ) = 1 1 is constant along solutions: −3 −2 −1 0 1 2 3 4 −1 � � H p ( t ) , q ( t ) = Const. −2 Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 4 / 43
Explicit Euler Implicit Euler q n +1 = q n + h p n q n +1 = q n + h p n +1 p n +1 = p n − h sin q n p n +1 = p n − h sin q n +1 2 2 1 1 −3 −2 −1 0 1 2 3 4 −3 −2 −1 0 1 2 3 4 −1 −1 −2 −2 Constant step size h = 0 . 3 in both cases. Initial values: large symbols. Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 5 / 43
Symplectic Euler methods q n +1 = q n + h p n q n +1 = q n + h p n +1 p n +1 = p n − h sin q n +1 p n +1 = p n − h sin q n 2 2 1 1 −3 −2 −1 0 1 2 3 4 −3 −2 −1 0 1 2 3 4 −1 −1 −2 −2 Constant step size h = 0 . 4 in both cases. Same initial values: large symbols. Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 6 / 43
Kepler problem (two-body problem) q i q i = p i , ˙ p i = − ˙ 2 ) 3 / 2 , i = 1 , 2 ( q 2 1 + q 2 exact solution in ( q 1 , q 2 )-space is an ellipse (drawn in red) 400 000 steps 1 h = 0 . 0005 −2 −1 1 −2 −1 1 4 000 steps −1 h = 0 . 05 explicit Euler symplectic Euler Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 7 / 43
N-body problems The differential equation is given by q i = ∂ H p i = − ∂ H � � ˙ p , q ) , ˙ p , q ) , i = 1 , . . . , N ∂ p i ∂ q i where i − 1 N N H ( p , q ) = 1 1 � p T � � � � i p i + V ij � q i − q j � 2 m i i =1 i =2 j =1 V ij ( r ) = − G m i m j Astronomy (solar system): r � � σ ij 12 � σ ij 6 � � � Molecular dynamics (Lennard–Jones): V ij ( r ) = ε ij − r r Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 8 / 43
Further examples of Hamiltonian systems Integrable Hamiltonian systems ◮ Toda lattice ◮ discretized Schr¨ odinger equation chaotic systems ◮ double (multiple) pendulum ◮ H´ enon–Heiles system rigid body dynamics ◮ free rigid body ◮ top, levitron Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 9 / 43
Lecture 1. Introduction to geometric integration Introduction and examples 1 Explicit, implicit, and symplectic Euler Kepler and N-body problems Hamiltonian systems 2 Symplectic mappings Theorem of Poincar´ e Symplectic Euler methods St¨ ormer–Verlet integrator Symplectic integration methods 3 Runge–Kutta methods Composition and splitting methods Variational integrators Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 10 / 43
Hamiltonian systems Differential equation: p i = − ∂ H q i = ∂ H � � � � ˙ p , q , ˙ p , q , i = 1 , . . . , d , ∂ q i ∂ p i where H : U → R and U ⊂ R d × R d . d . . . degree of freedom of the system. Different notation: p = −∇ q H ( p , q ) , ˙ q = ∇ p H ( p , q ) , ˙ or � 0 � p � � I y = J − 1 ∇ H ( y ) y = , J = . ˙ q − I 0 Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 11 / 43
Example: classical mechanical systems minimal coordinates . . . q ∈ R d q ) = 1 q T M ( q ) ˙ kinetic energy . . . T ( q , ˙ 2 ˙ q potential energy . . . U ( q ) � � � variational principle: T ( q , ˙ q ) − U ( q ) dt → min ∂ d � � � � Euler-Lagrange equ. M ( q ) ˙ q = T ( q , ˙ q ) − U ( q ) ∂ q dt momenta (or Poisson variables) . . . p := M ( q ) ˙ q Euler-Lagrange is equivalent to the Hamilton system with H ( p , q ) = 1 2 p T M ( q ) − 1 p + U ( q ) Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 12 / 43
Example: classical mechanical systems minimal coordinates . . . q ∈ R d q ) = 1 q T M ( q ) ˙ kinetic energy . . . T ( q , ˙ 2 ˙ q potential energy . . . U ( q ) � � � variational principle: T ( q , ˙ q ) − U ( q ) dt → min ∂ d � � � � Euler-Lagrange equ. M ( q ) ˙ q = T ( q , ˙ q ) − U ( q ) ∂ q dt momenta (or Poisson variables) . . . p := M ( q ) ˙ q Euler-Lagrange is equivalent to the Hamilton system with H ( p , q ) = 1 2 p T M ( q ) − 1 p + U ( q ) q = M ( q ) − 1 p = ∇ p H ( p , q ) Proof. ˙ p = 1 q ⊤ ∇ q M ( q ) ˙ ˙ 2 ˙ q − ∇ q U ( q ) = −∇ q H ( p , q ) Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 12 / 43
Symplectic mappings – differential form Let P be the parallelogram spanned by ξ = ( ξ p 1 , . . . , ξ p d , ξ q 1 , . . . , ξ q d ) T and η = ( η p 1 , . . . , η p d , η q 1 , . . . , η q d ) T and d d � ξ p η p � � � � � ξ p i η q i − ξ q i η p i i ω ( ξ, η ) := det = ξ q η q i i i i =1 i =1 the sum of the oriented areas of the projections of P onto the coordinate planes ( p i , q i ) . This bilinear form ω ( ξ, η ) can also be written as � 0 � I ω ( ξ, η ) := ξ T J η with J = − I 0 Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 13 / 43
Symplectic mappings – definition Definition A linear map A : R 2 d → R 2 d is symplectic if ω ( A ξ, A η ) = ω ( ξ, η ) for all ξ, η ∈ R 2 d or, equivalently, if A T J A = J . q q A η η A A ξ ξ p p Definition A differentiable map g : U → R 2 d (where U ⊂ R 2 d is an open set) is called symplectic if the Jacobian matrix g ′ ( p , q ) is everywhere symplectic, i.e., if g ′ ( p , q ) T J g ′ ( p , q ) = J . Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 14 / 43
Theorem of Poincar´ e We write the Hamiltonian system as y = J − 1 ∇ H ( y ) ˙ with H : U → R . (1) Theorem (Poincar´ e 1899) For every fixed t, the flow ϕ t of (1) is a symplectic transformation wherever it is defined. Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 15 / 43
Theorem of Poincar´ e We write the Hamiltonian system as y = J − 1 ∇ H ( y ) ˙ with H : U → R . (1) Theorem (Poincar´ e 1899) For every fixed t, the flow ϕ t of (1) is a symplectic transformation wherever it is defined. Proof. The derivative ∂ϕ t /∂ y 0 is a solution of the variational equation ˙ Ψ = J − 1 ∇ 2 H � � ϕ t ( y 0 ) Ψ. Hence, �� ∂ϕ t �� d � T � ∂ϕ t J = . . . = 0 . dt ∂ y 0 ∂ y 0 � ∂ϕ t � T � ∂ϕ t � Since J = J is satisfied for t = 0 ∂ y 0 ∂ y 0 ( ϕ 0 is the identity map), it is satisfied for all t and all y 0 . Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 15 / 43
Symplecticity is characteristic for Hamiltonian systems Theorem The flow ϕ t of y = f ( y ) is a symplectic transformation for all t if and ˙ only if locally f ( y ) = J − 1 ∇ H ( y ) for some H ( y ) . Ernst Hairer (Universit´ e de Gen` eve) Geometric Numerical Integration September 10 -14, 2018 16 / 43
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