3469 - Practical Numerical Simulations
Mike Peardon — mjp@maths.tcd.ie
http://www.maths.tcd.ie/~mjp/3469
School of Mathematics
Trinity College Dublin
Michaelmas Term 2015
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
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Numerical solutions to
hamiltonian dynamics
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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Solving Hamiltonian dynamics
Many standard integrators (Euler, Runge-Kutta) are unstable
when used to evolve equations of motion:
ẋ = p,
ṗ = f (x, t)
Example: Euler applied to simple harmonic oscillator;
xk+1
1 h
xk
=
pk+1
−h 1
pk
This means
k xk
1 h
x0
=
pk
−h 1
p0
and an analytic form for xk can be constructed.
Matrix eigenvalues are λ± = 1 ± ih, so |λ| > 1 when h > 0.
The amplitude of oscillations grows exponentially,
in
µt
2
proportion to e with µ = ln 1 + h
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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Symplectic integrators and leap-frog
A class of integrators for hamiltonian evolution can be
constructed with better properties - the symplectic
integrators
Simplest (reversible) example - leap-frog
The leap-frog integrator
1
2
3
xk+1/2 = xk + h2 pk
pk+1 = pk + hf (xk+1/2 , t + h/2)
xk+1 = xk+1/2 + h2 pk+1
This scheme is reversible: {xk , pk } → {xk+1 , pk+1 } means
{xk+1 , −pk+1 } → {xk , −pk }
Because of reversibility, leap-frog is a second-order scheme
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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Leap-frog (2)
From the recursion, x is updated twice consecutively, so
xk+3/2 = xk+1/2 + hpk+1
Now see where the method gets its name:
x0
p0
x 1/2
x 5/2
x 3/2
p1
p2
x3
p3
The leap-frog integrator
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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Leap-frog (3)
Leap-frog step for the SHO is a linear operator on {x, p}:
xk+1
1 0
xk
1 h2
1 h2
=
pk+1
−h 1
pk
0 1
0 1
2
3
1 − h2 h − h4
xk
=
h2
pk
−h
1− 2
Again, x k ∀k constructed from x k = M k x 0
Eigenvalues of M are
√
h2 ih 4 − h2
λ± = 1 −
±
2
2
Now for h ≤ 2, have |λ| = 1. For h > 2 at least one eigenvalue
has modulus greater than unity
Integrator is stable (solution stays bounded) for h < 2 for
SHO
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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Leap-frog (4)
2
Euler, h=0.1
Leap-frog, h=0.1
x(t)
1
0
-1
-2
0
2
4
6
8
10
12
14
16
18
20
t
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
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