on the extension of adams bashforth moulton methods for...
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On the Extension of Adams–Bashforth–MoultonMethods for Numerical Integration of Delay
Differential Equations and Applicationto the Moon’s Orbit
Dan Aksim and Dmitry Pavlov
Laboratory of Ephemeris AstronomyInstitute of Applied Astronomy of the Russian Academy of Sciences
St. Petersburg, Russia
October 9, 2019
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Types of differential equations
Ordinary differential equation (ODE):
x(t) = f(t,x(t))
(Retarded) delay differential equation (DDE):
x(t) = f(t,x(ϕ(t)), ...), ϕ(t) < t
Advanced differential equation:
x(t) = f(t,x(ψ(t)), ...), ψ(t) > t
DDE of neutral type:
x(t) = f(t, x(ξ(t)), ...), ξ(t) 6= t
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 2 / 14
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From retarded to advanced equations
Past Futuret
Epoch
Forwardintegration
0 1
x(t− τ)
Backwardintegration1 0
x(t+ τ)
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 3 / 14
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Rotational equation of the Moon (general form)
Forward: retarded DDE of neutral type with constant delays
x(t) = f(t,x(t),x(t− τ), x(t− τ))
Backward: advanced DDE of neutral type with constant delays
x(t) = f(t,x(t),x(t+ τ), x(t+ τ))
Initial condition at the epoch:
x(t0) = x0
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 4 / 14
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Adams–Bashforth–Moulton methods (I)
tn−r+1 tn tn+1
Lagrange polynomial
Integration
Interpolation (predictor)
Interpolation (corrector)
t
x
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 5 / 14
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Adams–Bashforth–Moulton methods (II)
1. Predictor — Adams–Bashforth (order 2):
xn+2 = xn+1 + h
(3
2fn+1 −
1
2fn
)2. Evaluation of fn+2 = f(tn+2,xn+2)
3. Corrector — Adams–Moulton (order 3):
xn+2 = xn+1 + h
(5
12fn+2 +
2
3fn+1 −
1
12fn
)4. (Optional). PECE, PECEC, PECECE
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 6 / 14
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Adams–Bashforth–Moulton methods (III)
Runge–Kutta jump start ABM
0 1 2 3 4
First (r − 1) steps must be performed by a single-step method.
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 7 / 14
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The ‘nested RK4’ method for DDEs
x(t) = f(t,x(t),x(t± τ), x(t± τ))x(t0) = x0
1. Introduce a new function
g(t) = f(t,x(t),x(t), x(t))
2. Retrieve delayed states by integrating g(t) with RK4
x(t± τ) = RK4It→t±τg(t)x(t± τ) = g(t± τ)
DrawbacksI Calculation of each delayed state requires 4 RHS calls
I No previous knowledge of x(t) is being used
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 8 / 14
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The interpolation method for DDEs (I)
tn − τ tn tn + τ
x(tn − τ)
x(tn + τ)
t
x
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 9 / 14
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The interpolation method for DDEs (II)
The algorithm
1. Jump start by the ‘nested RK4’ algorithm (with 8th orderDormand–Prince)
2. P and C stages are simple ABM
3. Each E stage constructs a Lagrange interpolating polynomial(any order), which is used to find x(t± τ) and x(t± τ)
Advantages
I Much cheaper delayed states
I No simplifying assumptions about the function
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 10 / 14
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Results. The forward–backward test
1984 1994 2004 2014
0
0.2
0.4
0.6
0.8
1
1.2
t
∆, mm
M position E–M distance Angular displacement
Orders: ABM — 13, interpolation and extrapolation — 9
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 11 / 14
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Results. The old–new test
1970 1980 1990 2000 2010 2018
0
1
2
3
t
∆, cm
M position E–M distance Angular displacement
Orders: ABM — 13, interpolation and extrapolation — 9
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 12 / 14
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Results. Approximation accuracy
1973 1979 1984 1990 1995 2001 2006 2012 2017
10−4
10−3
10−2
10−1
t
∆, m
Runge–KuttaInterpolation
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 13 / 14
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Results. O−C
Station Timespan NPs One-wayWRMS, cm
McDonald 1969–1985 3552/52 19.9
MLRS1 1983–1988 588/43 11.2
MLRS2 1988–2015 3216/454 3.4
Haleakala 1984–1990 748/22 5.7
OCA (Ruby) 1984–1986 1109/79 17.0
OCA (YAG) 1987–2005 8203/121 2.0
OCA (MeO) 2009–2017 1814/22 1.42
OCA (IR) 2015–2017 1814/20Old: 1.27
New: 1.26
APOLLO 2006–2016 2609/39 1.37
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 14 / 14
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Backup slides
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Tidal forces (I)
Earth M0
Moon
Earth M1
Low tide
High tide
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 16 / 14
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Tidal forces (II)
Earth
Moon
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 17 / 14
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Rotational equation of the Moon (actual form)
Euler’s equation for a rotating reference frame:
ω =
(I
m
)−1 [Nm− I
mω − ω ×
(I
mω
)],
ω — angular velocity,N(t) — torque,I/m — inertia tensor
I
m=I0m− Icm− k2
µE
µM
(RM
r
)5 x2 − 1
3r2 xy xz
xy y2 − 13r
2 yzxz yz z2 − 1
3r2
+ k2
R5M
3µM
ω2x − 1
3 (ω2 − n2) ωxωy ωxωz
ωxωy ω2y − 1
3 (ω2 − n2) ωyωz
ωxωz ωyωz ω2z − 1
3 (ω2 + 2n2)
,where r = (x, y, z)T = r(t− τ), ω = ω(t− τ), τ = 0.096 d
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 18 / 14
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Runge–Kutta methods
General form for the Runge–Kutta family of methods:
xn+1 = xn + h
s∑i=1
biki
ks = f(tn + csh,xn + h
s−1∑j=1
as,jkj)
DrawbacksI Butcher barriers:
• p ≥ 5: no RK method exists of order p with s = p stages• p ≥ 7: no RK method exists of order p with s = p+ 1 stages• p ≥ 8: no RK method exists of order p with s = p+ 2 stages
I Higher orders are problematic
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 19 / 14
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Other methods
1. Pavlov, IAA RAS
Nested Runge–Kutta method (described above)
2. Hofmann, Institut fur Erdmessung
Quadratic Taylor expansion using x, x and x
3. Williams, NASA JPL
Approximation with pre-fit polynomials (exact algorithmunknown)
Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 20 / 14