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Geometric Integration of Differential Equations
1. Introduction and ODEs
Chris Budd
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Want to simulate a physical system governed
by differential equations
Expect the numerical approximation to have the
same qualitative features as the underlying solution
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Traditional approach
Carefully approximate the differential operators in the system Solve the resulting difference equations
Monitor and control the local error
Basis of most black box codes and gives excellent resultsover moderate computing times
BUT This is a local process and does not pay attention to
the qualitative (global) features of the solution
Geometric Integration
Aims to reproduce qualitative and global features
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Some global features
Some qualitative properties:
Conservation laws
Global quantities: Energy, momentum, angular momentum
Flow invariants: Potential vorticity, Casimir functions
Phase space geometry
Symplectic structure
Symmetries
Galilean
Reversal
Scaling: Nonlinear Schrodinger
Lie Group: SO3 (Rigid body)
Asymptotic behaviour
Orderings
Often linked: Noethers theorem for Lagrangian flows
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Conserved quantities:
Symmetries: Rotation, Reflexion, Time reversal, Scaling
Kepler's Third Law
Hamiltonian Angular Momentum
Example: The Kepler Problem
2/3222
2
2/3222
2
)(,
)( yx
y
dt
yd
yx
x
dt
xd
!
yuxvLyx
vuH !
! ,)(
)(2
12/122
22 Q
),(),(),,(),(, 3/13/2 vuvuyxyxtt ppp PPP
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Geometric Integration
Aims to preserve a subset of these features
Take advantage of powerful global error estimates(shadowing)
Powerful methods for important physical problems
Examples of GI methods
Symplectic and multi-symplectic
Splitting
Lie Group/Magnus
Discrete LagrangianScale Invariant
Examples of GI applications
Molecular and celestial mechanics
Rigid body mechanics
Weather forecasting
Integrable systems (optics)Self-similar PDEs
Highly oscillatory problems
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Example of the traditional and the GI approach:
Integrating the Harmonic Oscillator
Qualitative features: Bounded periodic solutions, time reversal symmetry,
Conserved
Forward Euler method (non GI)
xdt
dyy
dt
dx!! ,
)(2
1 22 yxH !
nnnnnn hXYYhYXX !! 11 ,
nnnnn HhHYXH )1()(
2
1 21
22 !!
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Problem: Energy increases, lack of periodicity, lack of symmetry
Backward Euler Method (non GI)
Problem: Energy decreases, lack of periodicity, lack of symmetry
Mid-point rule (a GI method)
1111 , !! nnnnnn hXYYhYXX
)1/(2
1 hHH nn !
)(2
),(2
1111 !! nnnnnnnn XXh
YYYYh
XX
nn HH !1
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FE BE
Mid-Point rule
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Mid point rule conserves:
Energy
Symmetry
Backward (Modified) Equation Analysis
Solutions are: Bounded, periodic
Phase error proportional to
Discrete equation has an exact solution
Discrete solution shadows the continuous one
)(12
1),sin(),cos( 42
hOh
tYtX nnnn !!! [[[
th2
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Symplectic Methods
The mid-point rule behaves well because it conserves the
symplectic structure of the system
Classical Hamiltonian ordinary differential equation:
p
H
dt
dq
q
H
dt
dp
xx
!xx
! ,
!!
00,1
IIJHJ
dtdu),( qpu !
Differential equation induces a FLOW )(ut]
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FLOW MAP is symplectic
JJT !'' ]]
Symplecticity places a strong constraint on the flows
1. Preservation of phase space volume (and wedge product)
2. Recurrence
3. No evolution on a low dimensional attractor
4. KAM behaviour for near integrable systems
5. Composition of two symplectic flows is a symplectic flow
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Numerical method applied with a constant step size h gives a
map
)( phh hO!= ]
h=
Traditional analysis:
Show that
GI approach:
Show that is a symplectic map (symplectic method)h=
Advantage: Symplectic methods have good ergodic properties
Strong error estimates via backward error analysis
method is exact solution of a perturbed Hamiltonian problem
Symplectic Methods include: Runge-Kutta, Splitting, Variational
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Runge-Kutta methods for du/dt = f(u)
There is a large class of implicit symplectic Runge-Kutta methods
c A
b
Construct matrix M jijijijiij bbababM !Method is symplectic if M = 0
ButcherTableaux
All linearand quadratic invariants conserved
Luu
T
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)2/)(( 11 ! nnnn uuhfuu
Example: the implicit mid-point rule
All Gauss-Legendre Runge-Kutta methods and associated
collocation methods are symplectic
Symplectic, implicit, symmetric, unconditionally stable,
Conserves linear and quadratic invariants
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Splitting and composition methods
Runge-Kutta methods are implicit, but for certain problems we
can construct explicit symplectic methods via splitting
)()( 21 ufufdt
du
!
h,1= h,2=Construct flow maps and for and1f 2f
hhhT ,2,1, ==!= 2/,1.22/,1, hhhhS ===!=
Compose the split maps
Strang splittingLie-T
rotter splitting
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Some important results
If and are symplectic, so are
The Campbell-Baker-Hausdorff theorem implies that
h,1= h,2= hS,=hT,=
)( 2, hOhTh =!] )(3
, hOhSh =!]
If H(u) = T(p) + V(q)
The splittings lead directly to two important numerical methods
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)(),( 111 !! nnnnnn pThqqqVhpp
)(2
),(),(2
12/112/112/1 !!! nnnnnnnnn qVh
pppThqqqVh
pp
Symplectic Euler SE
Stormer-Verlet SV (Leapfrog)
Symplectic, explicit, non-symmetric, order 1
Symplectic, explicit, symmetric, order 2
Unstable for large step size
There are higher order, explicit, splitting methods due to Yoshida,
Blanes.
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Apply to the Kepler problem
SV
SE
FE
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Global error
H error
FE
SE
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Method Global error H error L error
FE t^2 h t h t h
SE t h h 0
SV t h^2 h^2 0
NOTE: Keplers third law is NOT conserved by these methods
see the next talk!
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Backward Error Analysis
Up to an exponentially small (In h) error
the solutions of a symplectic method oforder p are the
discrete samples of a solution of a related Hamiltonian
differential equation with Hamiltonian
...)()()()( 11
!
uHhuHhuHuHp
p
p
p
h
Can construct the perturbed Hamiltonian explicitly
H error remains bounded for all times
Doesnt apply if h varies!
)( /*hh
eOE
!
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Example:A problem in structural mechanics
22
12
qp
H !
)(12
12
)( 22
22
2
1 hOq
pqhq
phOhHHHh
!!
Discrete Euler Beam
Small h limit
Hamiltonian forSymplectic Euler discretisation = original problem
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hH
1hHH
h = 0.05 h = 1.1 h = 2.2
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Symmetry Group Methods
Important class of GI methods are used to solve problems
with Lie Group Symmetries (deep conservation laws)
gAGuuutAdt
du
! ,,),(
G: (matrix) Lie group g: Lie algebra
Eg. G = SO3 (rotations), g = so3 (skew symmetry)
Spo
tty
dog
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Can a numerical method ensure that the solution remains in G?
Rigid body mechanics, weather forecasting, quantum mechanics, Lyapunov
exponents, QR factorisation
Idea: Do all computations in the Lie Algebra (linear space)
And map between this and the Lie Group (nonlinear space)
G
g
nU 1nU
nW1n
Wnumerical method
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)2/()2/()(,!)exp(
1
AIAIAcayn
A
A
n
|!|!
WW
Examples of maps from g to G
General g , G g = so3, G = SO3
satisfies the dexpinv equation
),(!
n
ll UetAadl
B
dt
d WW
W!
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Integrate the dexpinv equation numerically
Conserve the group structure by making sure that allnumerical approximations to the dexpinv equation always
lie in the Lie algebra
Fine provided method uses linear operations and commutators
Runge-Kutta/Munthe-Kaas (RKMK) methods use this approach
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!tt
ddAAdAt0 0
21120
1
)](),([2
1)()(
\\\\\\\W
...)]()],(),([[4
132
0 0 01123
1 2
\\\\\\\ \
dddAAAt
utAdt
du)(!
Magnus series methods:
Magnus series:
Obtain method by series truncation and careful calculation of the
commutators
VERY effective for Highly Oscillatory Problems [Iserles]
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3
02/2/)cos(
2/0
2/)cos(0
)( so
tt
tt
tt
tA
!
Eg. Evolution on the surface of the sphere
uuIT
!invariant
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FE RK
RKMK Magnus
