# Asymptotic solvers for ordinary differential equations with multiple 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

## Summary (2 min read)

### 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.

### 3.6 Comments

- 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.

Did you find this useful? Give us your feedback

...read more

##### Citations

3 citations

### Cites methods from "Asymptotic solvers for ordinary dif..."

...We impose the same natural partial ordering on Ur as in [4]: first the singletons in lexicographic ordering, second the pairs in lexicographic ordering, then the triplets etc....

[...]

1 citations

##### References

1,245 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)....

[...]

1,215 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)....

[...]

742 citations

78 citations

### Additional excerpts

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

[...]

69 citations