Journal ArticleDOI

# Asymptotic solvers for ordinary differential equations with multiple frequencies

09 Sep 2015-Science China-mathematics (Science China Press)-Vol. 58, Iss: 11, pp 2279-2300
TL;DR: In this paper, the authors construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies.
Abstract: We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question Numerical examples illustrate the effectiveness of the method

### 1 Introduction

• The authors assume that at least some of these frequencies are large, thereby causing the solution to oscillate and rendering numerical discretization of (1.1) by classical methods expensive and inefficient.
• Cd is smooth, which has been already analysed at some length in (Condon, Deaño and Iserles 2010).
• Especially, the asymptotic expansion with a fixed number of terms becomes more accurate when increasing the oscillator parameter ω.
• The functions pr,m, which are all independent of ω, are constructed explicitly in a recursive manner.

### 2 The linear case

• (2.2) However, the finer structure of the solution is not apparent from (2.2) without some extra work.
• Each of the integrals hides an entire hierarchy of scales, and this becomes apparent once the authors expand them asymptotically.
• Note that linearity ‘locks’ frequencies: each pr,m depends just on am, m ∈ U0.

### 3.1 The recurrence relations

• There is a measure of redundancy in the last expression: for example there are two terms in Io2,3, namely (1,2) and (2,1), but they produce identical expressions.
• The authors may lump them together, paying careful attention to their multiplicity.
• Similarly to the expansion of (1.2) in (Condon et al. 2010), the authors obtain non-oscillatory differential equations for the coefficients pr,0, r ∈ Z+, which require initial conditions.

### 3.2 The first few values of r

• The expansion (1.7) exhibits two distinct hierarchies of scales: both amplitudes ω−r for r ∈ Z+ and, for each r ∈ N, frequencies eiσmωt.
• In that case the outcome would not have been consistent with the initial condition (3.5) and this is the rationale for the addition of this term.

### 3.3 The general case r ≥ 1

• The authors consider the ‘level r’ equations (3.4) noting that, by induction, the sets.
• The authors impose natural partial ordering on Ur: first the singletons in lexicographic ordering, then the pairs in lexicographic ordering, then the triplets etc.

### 3.5 Two non-commensurate frequencies

• This simplifies the argument a great deal.
• Greater, but not insurmountable effort is required to develop a general asymptotic expansion in this case.

• This is different from κms summing up exactly to zero: as demonstrated in Subsection 3.4, the authors can deal with the latter problem but not with the denominators in (3.7) or (3.8) becoming arbitrarily small in magnitude.
• Having said so, and bearing in mind that the truncation of (1.7) to r ≤ R yields an error of O(ω−R−1) and, |ω| being large, the authors are likely to obtain very high accuracy before small denominators kick in.
• Incidentally, this is precisely the reason for the requirement that, unlike in (1.2), the number of initial frequencies is finite.
• Unfortunately, this approach has another, more critical, shortcoming.
• The authors must identify all linear combinations of κms that sum up to zero, because they require an altogether different treatment.

### 4 Numerical experiments

• In the current section the authors present two examples that illustrate the construction of their expansions and demonstrate the effectiveness of their approach.
• In each case the authors compare the pointwise error incurred by a truncated expansion (1.7) with either the exact solution or the Maple routine rkf45 with exceedingly high error tolerance, using 20 significant decimal digits.

### 4.1 A linear example

• This being a linear equation, the exact solution and its asymptotic expansion are available explicitly using the theory from Section 2.
• It is clear that each time the authors increase s, the error indeed decreases substantially, in line with their theory.
• A comparison with the two previous figures emphasises the important point that the efficiency of the asymptotic - numerical method grows with ω, while the cost is to all intents and purposes identical.
• Indeed, wishing to produce similar error to their method with s = 4, the Maple routine rkf45 needs be applied with absolute and relative error tolerances of 10−13 and 10−18 respectively.

### 4.2 A nonlinear example in Memristor circuits

• The method in this paper is developed for Memristor circuits subject to high-frequency signals.
• The circuit 2The fact that Figs 4.1a and 4.1b are identical is a fluke-anyway, it is evident from the results on Page 5.
• 3Such error tolerances are impossible in Matlab, which explains their use of Maple.
• Then the asymptotic method is developed for this type of equation.

### 4.2.3 when r = 2

• Note that the (1,2) term and (3,4) terms are not present as addition of these frequencies would result in zero which is present in the set.
• Now consider the set U3 and the recursions for p3,m.

### 4.2.5 Numerical experiments

• The nonlinear Memristor circuits do not have a known analytical solution, the authors come to a reference solution, the Maple routine rkf45 with the accuracy tolerance AbsErr = 10−10 and RelErr = 10−10.
• Furthermore, with increasing the oscillatory parameter, the error of the asymptotic method decreases rapidly, a very important virtue of the method.
• In addition, the CPU time is compared to the Runge-Kutta method (rkf45) whose tolerance equals to 10−10.

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Asymptotic solvers for ordinary
diﬀerential equations with multiple
frequencies
Marissa CONDON
School of Electronic Engineering, Dublin City University
E-mail: marissa.condon@dcu.ie
Alfredo DEA
˜
NO
E-mail: alfredo.deanho@uc3m.es
Jing GAO
School of Mathematics and Statistics, Xi’an Jiaotong University
E-mail: jgao@mail.xjtu.edu.cn
Arieh ISERLES
DAMTP, Centre for Mathematical Sciences, University of Cambridge
E-mail: A.Iserles@damtp.cam.ac.uk
Abstract
We construct asymptotic expansions for ordinary diﬀerential equa-
tions with highly oscillatory forcing terms, focussing on the case of
multiple, non-commensurate frequencies. We derive an asymptotic ex-
pansion in inverse powers of the oscillatory parameter and use its trun-
cation as an exceedingly eﬀective means to discretize the diﬀerential
equation in question. Numerical examples illustrate the eﬀectiveness
of the method.
Mathematics Subject Classiﬁcation: 65L05,65D30,42B20,42A10
Communicating author
1

1 Introduction
Our concern in this paper is with the numerical solution of highly oscil-
latory ordinary diﬀerential equations of the form
y
0
(t) = f(y(t)) +
M
X
m=1
a
m
(t)e
m
t
, t 0, y(0) = y
0
C
d
, (1.1)
where f : C
d
C
d
and a
1
, ··· , a
M
: R
+
C
d
are analytic and ω
1
, ω
2
, ··· , ω
M
R \ {0} are given frequencies. We assume that at least some of these fre-
quencies are large, thereby causing the solution to oscillate and rendering
numerical discretization of (1.1) by classical methods expensive and ineﬃ-
cient. Many phenomena in engineer and physics are described by the oscil-
latoryly diﬀerential equations(Chedjou2001, Fodjouong2007, Slight2008 and
so on).
A special case of (1.1) with ω
2m1
= , ω
2m
= , m = 0, 1, ··· , bM/2c,
where ω 1, is a special case of
y
0
(t) = f(y(t)) + G(y)
X
k=−∞
b
k
(t)e
ikωt
, t 0, y(0) = y
0
C
d
, (1.2)
where G : C
d
× C
d
C
d
is smooth, which has been already analysed at
some length in (Condon, Dea˜no and Iserles 2010). It has been proved that
the solution of (1.2) can be expanded asymptotically in ω
1
,
y(t) p
0,0
(t) +
X
r=1
1
ω
r
X
m=−∞
p
r,m
(t)e
imωt
, t 0, (1.3)
where the functions p
r,m
, which are independent of ω, can be derived recur-
sively: p
r,0
by solving a non-oscillatory ODE and p
r,m
, m 6= 0, by recursion.
An alternative approach, based upon the Heterogeneous Multiscale Method
(E and Engquist 2003), is due to Sanz-Serna (2009). Although the theory
in (Sanz-Serna 2009) is presented for a speciﬁc equation, it can be extended
in a fairly transparent manner to (1.2) and, indeed, to (1.1). It produces
the solution in the form
y(t)
X
m=−∞
κ
m
(t)e
imωt
, (1.4)
2

where κ
m
(t) = O(ω
1
), m Z.
1
Formally, (1.3) and (1.4) are linked by
κ
m
(t) =
P
r=0
1
ω
r
p
r,0
(t), m = 0,
P
r=1
1
ω
r
p
r,m
(t), m 6= 0.
We adopt here the approach of (Condon et al. 2010), because it allows us
to derive the expansion in a more explicit form.
The highly oscillatory term in (1.2) is periodic in : the main diﬀerence
with our model (1.1) is that we allow the more general setting of almost
periodic terms (Besicovitch 1932). It is justiﬁed by important applications,
not least in the modelling of nonlinear circuits (Giannini and Leuzzi 2004,
Ram´ırez, Su´arez, Lizarraga and Collantes 2010).
Another diﬀerence is that we allow in (1.1) only a ﬁnite number of dis-
tinct frequencies in the forcing term. This is intended to prevent the oc-
currence of small denominators, familiar from asymptotic theory (Verhulst
1990). Note that (Chartier, Murua and Sanz-Serna 2012) (cf. also (Chartier,
Murua and Sanz-Serna 2010)) employs similar formalism - a ﬁnite number
of multiple, noncommensurate frequencies - except that it does so within
the ‘body’ of the diﬀerential operator, rather than in the forcing term.
We commence our analysis by letting U
0
= {1, 2, ··· , M} and ω
j
= κ
j
ω,
j = 1, ··· , M, where ω is a large number which will serve as our asymptotic
parameter. Consequently, we can rewrite (1.1) in a form that emphasises
the similarities and identiﬁes the diﬀerences with (1.2),
y
0
(t) = f(y(t)) +
X
m∈U
0
a
m
(t)e
m
ωt
, t 0, y(0) = y
0
C
d
, (1.5)
Section 2 is devoted to a ‘warm up exercise’, an asymptotic expansion of a
linear version of (1.5), namely
y
0
(t) = Ay(t) +
X
m∈U
0
a
m
(t)e
m
ωt
, t 0, y(0) = y
0
C
d
, (1.6)
where A is a d × d matrix. Of course, the solution of (1.5) can be written
explicitly, but this provides little insight into the real size of diﬀerent com-
ponents. The asymptotic expansion is considerably more illuminating, as
1
In the special case considered in (Sanz-Serna 2009) it is true that κ
m
(t) = O (ω
−|m|
),
m Z, but this does not generalise to (1.2).
3

well as hinting at the general pattern which we might expect once we turn
our gaze to the nonlinear equation (1.5).
For the ODE with the highly oscillatory forcing terms with multiple
frequencies, the asymptotic method is superior to the standard numerical
methods. With less computational expense, this asymptotic method can
obtain the higher accuracy. Especially, the asymptotic expansion with a
ﬁxed number of terms becomes more accurate when increasing the oscillator
parameter ω.
In Section 3 we will demonstrate the existence of sets
U
0
U
1
U
2
···
and of a mapping
σ :
[
r=0
U
r
R
such that the solution of (1.5) can be written in the form
y(t) p
0,0
(t) +
X
r=1
1
ω
r
X
m∈U
r
p
r,m
(t)e
m
ωt
, t 0. (1.7)
As can be expected, the original parameters {κ
1
, κ
2
, ··· , κ
M
} form a subset
of U
r
. However, we will see in the sequel that the set σ(U
r
) is substantially
larger for r 3. In the sequel we refer to elements of σ(U
r
) as {σ
m
: m U
r
}.
The functions p
r,m
, which are all independent of ω, are constructed
explicitly in a recursive manner. We will demonstrate that the sets U
r
are
composed of n-tuples of nonnegative integers.
In Section 4 we accompany our narrative by a number of computational
results. Setting the error functions are represented as
s
(t, ω) = y(t) p
0,0
(t)
s
X
r=1
1
ω
r
X
m∈U
r
p
r,m
(t)e
m
ωt
,
we plot the error functions in the ﬁgures to illustrate the theoretical analysis.
The expansion solvers are convergent asymptotically. That is, for every
ε > 0, ﬁxed s, the bounded interval for t, there exists ω
0
> 0 such that for
ω > ω
0
, the error function |
s
(t, ω)| < ε. However, for increasing s, ﬁxed t
and ω, we will come to the convergence of the expansion in future paper.
4

2 The linear case
Our concern in this section is with the linear highly oscillatory ODE
(1.6), which we recall for convenience,
y
0
= Ay +
X
m∈U
0
a
m
(t)e
m
ωt
, t 0, y(0) = y
0
C
d
, (2.1)
Its closed-form solution can be derived at once from standard variation of
constants,
y(t) = e
tA
y
0
+ e
tA
X
m∈U
0
Z
t
0
e
xA
a
m
(x)e
m
ωx
dx. (2.2)
However, the ﬁner structure of the solution is not apparent from (2.2) with-
out some extra work. Each of the integrals hides an entire hierarchy of
scales, and this becomes apparent once we expand them asymptotically.
The asymptotic expansion of integrals with simple exponential operators
is well known: given g C
[0, t) and |η| 1,
Z
t
0
g(x)e
x
dx
X
r=1
1
()
r
h
g
(r1)
(t)e
t
g
(r1)
(0)
i
(Iserles, Nørsett and Olver 2006). For any m U
0
we thus take g(x) =
e
xA
a
m
(x) and η = κ
m
ω (recall that κ
m
6= 0), therefore
y(t) e
tA
y
0
X
m∈U
0
X
r=1
1
(
m
ω)
r
"
e
m
ωt
r1
X
=0
(1)
r1
r 1

A
r1
a
()
m
(t)
e
tA
r1
X
=0
(1)
r1
r 1

A
r1
a
()
m
(0)
#
= e
tA
y
0
+
X
r=1
1
ω
r
X
m∈U
0
"
e
m
ωt
(
m
)
r
r1
X
=0
(1)

r 1

A
r1
a
()
m
(t)
1
(
m
)
r
e
tA
r1
X
=0
(1)

r 1

A
r1
a
()
m
(0)
#
.
We deduce the expansion (1.7), with U
m
= {0}U
0
= {0, 1, ··· , M}, m N,
and the coeﬃcients
p
0,0
(t) = e
tA
y
0
,
5

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01 Mar 2014-Calcolo
TL;DR: In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies, and its truncation results in a very effective method of dicretizing the differential equation system in question.
Abstract: In this paper, an asymptotic expansion is constructed to solve second-order differential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very effective method of dicretizing the differential equation system in question. Numerical experiments illustrate the effectiveness of the asymptotic method in contrast to the standard Runge---Kutta method.

3 citations

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##### References
More filters
Book
01 Jan 1990
TL;DR: In this article, the Poincare-Bendixson theorem is applied to the analysis of two-dimensional linear systems with first integrals and integral manifolds, and the Lagrange standard form is used.
Abstract: 1 Introduction.- 1.1 Definitions and notation.- 1.2 Existence and uniqueness.- 1.3 Gronwall's inequality.- 2 Autonomous equations.- 2.1 Phase-space, orbits.- 2.2 Critical points and linearisation.- 2.3 Periodic solutions.- 2.4 First integrals and integral manifolds.- 2.5 Evolution of a volume element, Liouville's theorem.- 2.6 Exercises.- 3 Critical points.- 3.1 Two-dimensional linear systems.- 3.2 Remarks on three-dimensional linear systems.- 3.3 Critical points of nonlinear equations.- 3.4 Exercises.- 4 Periodic solutions.- 4.1 Bendixson's criterion.- 4.2 Geometric auxiliaries, preparation for the Poincare-Bendixson theorem.- 4.3 The Poincare-Bendixson theorem.- 4.4 Applications of the Poincare-Bendixson theorem.- 4.5 Periodic solutions in ?n.- 4.6 Exercises.- 5 Introduction to the theory of stability.- 5.1 Simple examples.- 5.2 Stability of equilibrium solutions.- 5.3 Stability of periodic solutions.- 5.4 Linearisation.- 5.5 Exercises.- 6 Linear Equations.- 6.1 Equations with constant coefficients.- 6.2 Equations with coefficients which have a limit.- 6.3 Equations with periodic coefficients.- 6.4 Exercises.- 7 Stability by linearisation.- 7.1 Asymptotic stability of the trivial solution.- 7.2 Instability of the trivial solution.- 7.3 Stability of periodic solutions of autonomous equations.- 7.4 Exercises.- 8 Stability analysis by the direct method.- 8.1 Introduction.- 8.2 Lyapunov functions.- 8.3 Hamiltonian systems and systems with first integrals.- 8.4 Applications and examples.- 8.5 Exercises.- 9 Introduction to perturbation theory.- 9.1 Background and elementary examples.- 9.2 Basic material.- 9.3 Naive expansion.- 9.4 The Poincare expansion theorem.- 9.5 Exercises.- 10 The Poincare-Lindstedt method.- 10.1 Periodic solutions of autonomous second-order equations.- 10.2 Approximation of periodic solutions on arbitrary long time-scales.- 10.3 Periodic solutions of equations with forcing terms.- 10.4 The existence of periodic solutions.- 10.5 Exercises.- 11 The method of averaging.- 11.1 Introduction.- 11.2 The Lagrange standard form.- 11.3 Averaging in the periodic case.- 11.4 Averaging in the general case.- 11.5 Adiabatic invariants.- 11.6 Averaging over one angle, resonance manifolds.- 11.7 Averaging over more than one angle, an introduction.- 11.8 Periodic solutions.- 11.9 Exercises.- 12 Relaxation Oscillations.- 12.1 Introduction.- 12.2 Mechanical systems with large friction.- 12.3 The van der Pol-equation.- 12.4 The Volterra-Lotka equations.- 12.5 Exercises.- 13 Bifurcation Theory.- 13.1 Introduction.- 13.2 Normalisation.- 13.3 Averaging and normalisation.- 13.4 Centre manifolds.- 13.5 Bifurcation of equilibrium solutions and Hopf bifurcation.- 13.6 Exercises.- 14 Chaos.- 14.1 Introduction and historical context.- 14.2 The Lorenz-equations.- 14.3 Maps associated with the Lorenz-equations.- 14.4 One-dimensional dynamics.- 14.5 One-dimensional chaos: the quadratic map.- 14.6 One-dimensional chaos: the tent map.- 14.7 Fractal sets.- 14.8 Dynamical characterisations of fractal sets.- 14.9 Lyapunov exponents.- 14.10 Ideas and references to the literature.- 15 Hamiltonian systems.- 15.1 Introduction.- 15.2 A nonlinear example with two degrees of freedom.- 15.3 Birkhoff-normalisation.- 15.4 The phenomenon of recurrence.- 15.5 Periodic solutions.- 15.6 Invariant tori and chaos.- 15.7 The KAM theorem.- 15.8 Exercises.- Appendix 1: The Morse lemma.- Appendix 2: Linear periodic equations with a small parameter.- Appendix 3: Trigonometric formulas and averages.- Appendix 4: A sketch of Cotton's proof of the stable and unstable manifold theorem 3.3.- Appendix 5: Bifurcations of self-excited oscillations.- Appendix 6: Normal forms of Hamiltonian systems near equilibria.- Answers and hints to the exercises.- References.

1,290 citations

### "Asymptotic solvers for ordinary dif..." refers background in this paper

• ...Comment 1: As we increase levels r, we are increasingly likely to encounter the well-known phenomenon of small denominators (Verhulst 1990): unless κm = cψm, m = 1, · · · ,M , where all the ψms are rational (in which case, replacing ω by its product with the least common denominator of the ψms, we…...

[...]

• ...This is intended to prevent the occurrence of small denominators, familiar from asymptotic theory (Verhulst 1990)....

[...]

Book
01 Jan 1954

1,219 citations

### "Asymptotic solvers for ordinary dif..." refers background in this paper

• ...There is a measure of redundancy in the last expression: for example there are two terms in I2,3, namely (1,2) and (2,1), but they produce identical expressions....

[...]

• ...Since I1,2 = {2} and I2,2 = {(1, 1)}, we have from (3....

[...]

• ...Next to U3 = {0, 1, 2, 3, (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}, with σ1,1 = 2, σ1,2 = 1 + √ 2, σ1,3 = − √ 2, σ2,2 = 2 √ 2, σ2,3 = −1 and σ3,3 = −2− 2 √ 2....

[...]

• ...For example, in the case r = 3 we have I1,3 = {3}, I2,3 = {(1, 2)}, I3,3 = {(1, 1, 1)}, θ3 = θ1,1,1 = 1, θ1,2 = 2, while U3 = {0, 1, · · · ,M}∪{(m1,m2) : m1 ≤ m2, κm1+κm2 6= κm, ∀m = 0, · · · ,M}....

[...]

• ...…all the ψms are rational (in which case, replacing ω by its product with the least common denominator of the ψms, we are back to the case (1.2) of frequencies being integer multiples of ω) and positive, the set of all finite-length linear combinations of the κms is dense in R (Besicovitch 1932)....

[...]

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Abstract: The heterogenous multiscale method (HMM) is presented as a general methodology for the efficient numerical computation of problems with multiscales and multiphysics on multigrids. Both variational and dynamic problems are considered. The method relies on an efficent coupling between the macroscopic and microscopic models. In cases when the macroscopic model is not explicity available or invalid, the microscopic solver is used to supply the necessary data for the microscopic solver. Besides unifying several existing multiscale methods such as the ab initio molecular dynamics [13], quasicontinuum methods [73,69,68] and projective methods for systems with multiscales [34,35], HMM also provides a methodology for designing new methods for a large variety of multiscale problems. A framework is presented for the analysis of the stability and accuracy of HMM. Applications to problems such as homogenization, molecular dynamics, kinetic models and interfacial dynamics are discussed.

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78 citations

• ...It is justified by important applications, not least in the modelling of nonlinear circuits (Giannini and Leuzzi 2004, Ramı́rez, Suárez, Lizarraga and Collantes 2010)....

[...]

Book ChapterDOI
, S. Olver1
01 Jan 2006
TL;DR: Iserles et al. as discussed by the authors presented an analysis of the relationship between the two types of models and showed that Olver's model is more similar to the one proposed by Norsett and Serles.
Abstract: 1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, A.Iserles@damtp.cam.ac.uk 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway, S.P.Norsett@math.ntnu.no 3 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, S.Olver@damtp.cam.ac.uk

72 citations