Splitting methods in geometric numerical integration of differential - PowerPoint PPT Presentation

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Splitting methods in geometric numerical integration of differential equations Fernando Casas Fernando.Casas@mat.uji.es www.gicas.uji.es Departament de Matemàtiques Universitat Jaume I Castellón, Spain Barcelona, 3 December 2008

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Based on the paper Splitting and composition methods in the numerical integration of differential equations by Ander Murua Universidad del País Vasco San Sebastián, Spain Sergio Blanes Universidad Politécnica de Valencia Valencia, Spain and F . C. Supported by MEC (Spain), project MTM2007-61572

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis What is splitting ? Given the initial value problem x ′ = f ( x ) , x 0 = x ( 0 ) ∈ R D (1) with f : R D − → R D and solution ϕ t ( x 0 ) , suppose that m � f [ i ] : R D − f [ i ] , → R D f = i = 1 such that x ′ = f [ i ] ( x ) , x 0 = x ( 0 ) ∈ R D , i = 1 , . . . , m (2) can be integrated exactly, with solutions x ( h ) = ϕ [ i ] h ( x 0 ) at t = h . Then χ h = ϕ [ m ] ◦ · · · ◦ ϕ [ 2 ] h ◦ ϕ [ 1 ] (3) h h verifies χ h ( x 0 ) = ϕ h ( x 0 ) + O ( h 2 ) . First order approximation

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis What is splitting ? Three steps in splitting: choosing the set of functions f [ i ] such that f = � i f [ i ] 1 solving either exactly or approximately each equation 2 x ′ = f [ i ] ( x ) combining these solutions to construct an approximation for 3 x ′ = f ( x ) Remark: equations x ′ = f [ i ] ( x ) should be simpler to integrate than the original system.

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Some advantages of splitting methods Simple to implement. They are, in general, explicit. Their storage requirements are quite modest. They preserve structural properties of the exact solution: symplecticity, volume preservation, time-symmetry and conservation of first integrals Splitting methods constitute an important class of geometric numerical integrators Aim of geometric numerical integration: reproduce the qualitative features of the solution of the differential equation being discretised, in particular its geometric properties.

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis More on geometric integration Properties of the system are built into the numerical method. This gives the method an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods. Important aspect: theoretical explanation of the relationship between preservation of the geometric properties and the observed favourable error propagation in long-time integration (backward error analysis).

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 1: symplectic Euler and leapfrog q , p ∈ R d . Hamiltonian H ( q , p ) = T ( p ) + V ( q ) , Equations of motion: q ′ = T p ( p ) , p ′ = − V q ( q ) Euler method: q n + 1 = q n + hT p ( p n ) (4) = p n − hV q ( q n ) . p n + 1 H is the sum of two Hamiltonians, the first one depending only on p and the second only on q with equations q ′ q ′ = T p ( p ) = 0 and (5) p ′ p ′ = 0 = − V q ( q )

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 1: symplectic Euler and leapfrog q , p ∈ R d . Hamiltonian H ( q , p ) = T ( p ) + V ( q ) , Equations of motion: q ′ = T p ( p ) , p ′ = − V q ( q ) Euler method: q n + 1 = q n + hT p ( p n ) (4) = p n − hV q ( q n ) . p n + 1 H is the sum of two Hamiltonians, the first one depending only on p and the second only on q with equations q ′ q ′ = T p ( p ) = 0 and (5) p ′ p ′ = 0 = − V q ( q )

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 1: symplectic Euler and leapfrog q , p ∈ R d . Hamiltonian H ( q , p ) = T ( p ) + V ( q ) , Equations of motion: q ′ = T p ( p ) , p ′ = − V q ( q ) Euler method: q n + 1 = q n + hT p ( p n ) (4) = p n − hV q ( q n ) . p n + 1 H is the sum of two Hamiltonians, the first one depending only on p and the second only on q with equations q ′ q ′ = T p ( p ) = 0 and (5) p ′ p ′ = 0 = − V q ( q )

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 1: symplectic Euler and leapfrog Solution: q ( t ) = q 0 + t T p ( p 0 ) ϕ [ T ] : (6) t p ( t ) = p 0 q ( t ) = q 0 ϕ [ V ] : t p ( t ) = p 0 − t V q ( q 0 ) Composing the t = h flows gives the scheme p n + 1 = p n − h V q ( q n ) χ h ≡ ϕ [ T ] ◦ ϕ [ V ] : (7) h h q n + 1 = q n + h T p ( p n + 1 ) . χ h is a symplectic integrator, since it is the composition of flows of two Hamiltonians: symplectic Euler method

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 1: symplectic Euler and leapfrog By composing in the opposite order, ϕ [ V ] ◦ ϕ [ T ] h , another h first order symplectic Euler scheme: q n + 1 = q n + h T p ( p n ) h ≡ ϕ [ V ] ◦ ϕ [ T ] χ ∗ : (8) h h p n + 1 = p n − h V q ( q n + 1 ) . (8) is the adjoint of χ h . Another possibility: ‘symmetric’ version S [ 2 ] ≡ ϕ [ V ] h / 2 ◦ ϕ [ T ] ◦ ϕ [ V ] h / 2 , (9) h h Strang splitting, leapfrog or Störmer–Verlet method Observe that S [ 2 ] = χ h / 2 ◦ χ ∗ h / 2 and it is also symplectic h and second order.

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 2: Simple harmonic oscillator 2 ( p 2 + q 2 ) , where now q , p ∈ R . H ( q , p ) = 1 Equations: � q ′ � � 0 � � q � � � � � 1 0 0 x ′ ≡ = + = ( A + B ) x . p ′ 0 0 − 1 0 p � �� � � �� � A B Euler scheme: � q n + 1 � � q n � � � 1 h = , p n + 1 − h 1 p n Symplectic Euler method: � q n + 1 � � q n � q n � � � � 1 h = e hB e hA = . 1 − h 2 p n + 1 − h p n p n

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 2: Simple harmonic oscillator Both render first order approximations to the exact solution x ( t ) = e h ( A + B ) x 0 , but there are important differences Symplectic Euler is area preserving and 1 n + 1 ) = 1 2 ( p 2 n + 1 + hp n + 1 q n + 1 + q 2 2 ( p 2 n + hp n q n + q 2 n ) . Symplectic Euler is the exact solution at t = h of the perturbed Hamiltonian system 2 arcsin ( h / 2 ) ( p 2 + h p q + q 2 ) ˜ H ( q , p , h ) = √ (10) 4 − h 2 h � 1 � 1 2 p q + 1 2 ( p 2 + q 2 ) + h 12 h ( p 2 + q 2 ) + · · · = .

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 2: Simple harmonic oscillator How these features manifest in practice? Initial conditions ( q 0 , p 0 ) = ( 4 , 0 ) and integrate with a time step h = 0 . 1 (same computational cost) with Euler and symplectic Euler Two experiments: Represent the first 5 numerical approximations 1 Represent the first 100 points in the trajectory 2

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 2: Simple harmonic oscillator h=1/10 h=1/10 6 0 4 − 0.5 2 − 1 p p 0 − 2 − 1.5 − 4 − 2 − 6 3 3.2 3.4 3.6 3.8 4 4.2 − 6 − 4 − 2 0 2 4 6 q q Euler method (white circles) and the symplectic Euler method (black circles) with initial condition ( q 0 , p 0 ) = ( 4 , 0 ) and h = 0 . 1.

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 3: The 2-body (Kepler) problem Hamiltonian � H ( q , p ) = T ( p )+ V ( q ) = 1 2 ) − 1 2 ( p 2 1 + p 2 q 2 1 + q 2 r , r = 2 . Initial condition: � 1 + e q 1 ( 0 ) = 1 − e , q 2 ( 0 ) = 0 , p 1 ( 0 ) = 0 , p 2 ( 0 ) = 1 − e , 0 ≤ e < 1 is the eccentricity of the orbit. Total energy H = H 0 = − 1 / 2, period of the solution is 2 π . Two experiments with e = 0 . 6. We compare Euler and symplectic Euler

Introduction with examples Splitting and composition methods Order conditions of splitting and composition methods Backward error analysis Example 3: The 2-body (Kepler) problem 1.5 1.5 1 1 0.5 0.5 Y 0 Y 0 − 0.5 − 0.5 − 1 − 1 − 1.5 − 1.5 − 2.5 − 2 − 1.5 − 1 − 0.5 0 0.5 − 2.5 − 2 − 1.5 − 1 − 0.5 0 0.5 X X 1 The left panel shows the results for h = 100 and the first 3 1 periods and the right panel shows the results for h = 20 and the first 15 periods.