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

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

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

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

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

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

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

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

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

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

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

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

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

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

Backup slides

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

Tidal forces (II)

Earth

Moon

Dan Aksim and Dmitry Pavlov ABM, DDEs and the Moon’s Orbit IAA RAS 17 / 14

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

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

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