TWO-DIMENSIONAL LONG-TIME ASYMPTOTICS

C h a p t e r 3

differ in that for times𝑡 such that𝑡 −𝑡⁰ > |r−r⁰|/𝑐the three-dimensional Green’s function vanishes and the two-dimensional Green’s function does not. Indeed, the two-dimensional Green’s function in that region is a slowly-decreasing function of time𝑡, a fact that follows from Hadamard’s classical method of descent [68, §29]

for the wave equation, whereby the wave equation in two dimensions is viewed as a solution to the three-dimensional wave equation that is constant with respect to one spatial variable. As remarked in the above epigraph, voices from the past are heard infinitely far into the future.

As might be expected, differences of this nature persist in wave scattering prob- lems. Indeed, while for many three-dimensional scattering problems the scattering solutions decay rapidly (for much more discussion of known decay rates in three dimensions, see Section 13.2 of Chapter 4), the solution of the two-dimensional Dirichlet problem, for example, can decay as slowly as 1/(𝑡log²(𝑡)) [98]. (The asymptotics for the Neumann problem were first established in reference [4] and are included along with asymptotics for the Robin and impedance problems in [98, Thm. 6].) Furthermore, the difference in scattering character between the two- and three-dimensional problems is clearly evidenced by a certain result on the solution asymptotics presented in reference [98, Thm. 8], establishing that if a two- dimensional wave equation solution decays superalgebraically fast, it is identically zero (cf. the discussion in regards to three-dimensional decay rates in Section 13.2 showing superalgebraic or exponential decay rates for many classes of obstacle ge- ometries). In our context, the relevant long-time asymptotics to a solution𝑢of (1.6) in𝑑 =2 dimensions are given by [98, Thm. 6]

𝑢(r, 𝑡) ∼ 1 𝑡

∞

Õ

𝑗=2

𝑐0, 𝑗(r) log^𝑗(𝑡) +

∞

Õ

𝑞=1

1 𝑡2𝑞+1

∞

Õ

𝑗=1−𝑞 𝑗≠1

𝑐_{𝑞, 𝑗}(r) log^𝑗(𝑡)

, 𝑡 → ∞. (3.2)

This asymptotic expression carries important implications for the computational methodology proposed in this thesis. Indeed, in the proposed methodology, the solution𝑢is expressed (see (2.10)) as the sum

𝑢(r, 𝑡) =

𝐾

Õ

𝑘=1

𝑢_𝑘(r, 𝑡), (3.3)

where the functions𝑢_𝑘 are solutions to the wave equation (1.6) with Dirichlet data 𝑏 substituted by a windowed section 𝑏_𝑘 (supported in [𝑠_𝑘 − 𝐻 , 𝑠_𝑘 + 𝐻]) of the overall given incident field. The slow asymptotic decay of solutions 𝑢_𝑘 implies

that many of the 𝐾 = O (𝑇^inc) windows are important for solution accuracy at large times 𝑡. Indeed, due to the slow time-decay of solutions 𝑢_𝑘 resulting from incident fields separated by 𝐻 in time, truncation of the sum in (3.3) to a number of terms independent of 𝐾 would result in unacceptably large error for large 𝑡 as 𝐾 → ∞—since the series of general term Í

𝑘1/( (𝑡− 𝑠_𝑘)log²(𝑡 − 𝑠_𝑘)), while convergent, converges extremely slowly. Therefore, in order to avoid large errors as 𝑇^inc and 𝑡 grow, a number of terms proportional to 𝑇^inc must be kept in the sum, in stark contrast with the 𝑑 = 3 case for which often a finite and bounded number of terms accurately represent the solution for arbitrarily large values of 𝑡 and𝑇^inc. An interesting conclusion follows, that the “time-leaping” capability of the hybrid method to produce the full solution𝑢is only applicable in three-dimensional contexts, since, in two dimensions, solutions𝑢_𝑘 from incident fields in the distant past contribute in a significant manner to the overall solution𝑢 over vast stretches of time.

If the final solution time 𝑇 is proportional to the duration of the incident field, 𝑇 ∝𝑇^inc, then since the number of windows𝐾 =O (𝑇^inc), the evaluation of𝑢at time sample points of fixed spacing throughout [0, 𝑇]will requireO (𝑇 𝐾)evaluations of solution components𝑢_𝑘, precisely𝑇 for each 1≤ 𝑘 ≤ 𝐾. Indeed, while the methods of Chapter 2 may ensure that the cost to evaluate the field contribution𝑢_𝑘 from any single window with arbitrary𝑘 (1 ≤ 𝑘 ≤ 𝐾) at a single point in time is onlyO (1) with respect to𝑇, the fact that field contributions may be relevant for a long time is as yet unaddressed. This section proposes an algorithm to efficiently and accurately evaluate the total field𝑢over vast stretches of time while needing only to complete a total ofO (𝑇)evaluations of any of the𝑢_𝑘 solutions across all of 1 ≤ 𝑘 ≤ 𝐾. First, let 𝑡_𝑘 > 𝑠_𝑘 be a time at which the solution 𝑢_𝑘 has been identified to enter the asymptotic regime of (3.2) (recall from Chapter 2 that 𝑠_𝑘 is the time-center of partition 𝑘). The algorithm then fits a polynomial to 𝑢_𝑘 on [𝑡_𝑘,∞) (see also Remark 15), and, in service of this, we make certain manipulations to the asymptotic expansion to render it more amenable for fitting on an infinite interval. Thus, considering the change of variables𝑧_𝑘(𝑡)=1/log(𝑡−𝑠_𝑘), the asymptotic expansion of𝑢_𝑘 (Equation (3.2) with𝑢=𝑢_𝑘) becomes

𝑢_𝑘(r,e^1/𝑧𝑘 +𝑠_𝑘) ∼

∞

Õ

𝑗=2

𝑐0, 𝑗(r)𝑧

𝑗

𝑘e^{−1/𝑧𝑘} +

∞

Õ

𝑞=1

∞

Õ

𝑗=1−𝑞 𝑗≠1

𝑐_{𝑞, 𝑗}(r)𝑧

𝑗

𝑘e^{−(2𝑞+1)/𝑧𝑘}. (3.4) It is easy to see from the asymptotics that this is a smooth function of𝑧_𝑘 that can be easily fit with polynomials on the bounded interval [0, 𝑧_𝑘(𝑡_𝑘)]. Before doing this,

for all 𝑘, 𝑢_𝑘 is multiplied by e^1/𝑧𝑘 (a procedure that further smooths the function, as can be seen by consideration of (3.4)). Then, for 𝑘 = 1, at each spatial pointr, a polynomial 𝑃^r

𝑘(𝑧) = 𝑃^r

1(𝑧) with zero constant term (𝑃^r

1(0) =0) is obtained such that𝑃^r

1(𝑧_𝑘(𝑡))fits e^1/𝑧1𝑢

1(r,e^1/𝑧1 +𝑠

1), in the least-squares sense, over an𝑛_𝑠-point equidistant mesh in the interval [0, 𝑧

1(𝑣

1)]. The zero-vanishing assumption on the least-squares polynomial𝑃^r

1(𝑧) embodies the𝑢

1 → 0 behavior as𝑡 → +∞, which itself corresponds to𝑧

1(𝑡) →0⁺.

The algorithm then proceeds iteratively through the window indices 𝑘, so that for each 𝑘 > 1 and for a given spatial point r, 𝑃^r

𝑘(𝑧) approximates Í𝑘

ℓ=1𝑢_ℓ. More precisely, for 𝑘 > 1 and for each pointrin a selected spatial evaluation set R (cf.

Section 7 in Chapter 2), the polynomial𝑃^r

𝑘(𝑧)with zero constant term (𝑃^r

𝑘(0) =0) is obtained that fits e^1/𝑧𝑘𝑢_𝑘(r, 𝑠_𝑘 +e^{1/𝑧𝑘(𝑡)}) +𝑃^r

𝑘−1(𝑧) in a least squares sense over the𝑛_𝑠-point equidistant mesh in the interval [0, 𝑧_𝑘(𝑡_𝑘)]. Therefore, for𝑡 > 𝑡_𝑘, the approximation

𝑘

Õ

ℓ=1

𝑢_ℓ ≈ 𝑃^r

𝑘(𝑧_𝑘(𝑡))

holds, for𝑡 > 𝑡_𝑘, up to the accuracy of the least squares fit and the discretization error underlying the numerical evaluation of the solutions𝑢_𝑘. The overall solution can thus be efficiently obtained for all time by completing this procedure for 𝑘 =1, . . . , 𝐾. Clearly, the entire methodology is enabled by the capability to evaluate 𝑢_𝑘 on arbitrary time grids without reliance on preceding time values.

Remark 15. For simplicity, the description of the proposed asymptotic polynomial representation assumes use of a single asymptotic-matching polynomial. But use of multiple matching polynomials in the interval [0, 𝑧_𝑘(𝑡_𝑘)], one corresponding to an unbounded time interval the rest corresponding to bounded time intervals, may be advantageous in practice. For example, for the numerical experiments in Section11 two polynomials of degree five were used for asymptotic approximation within the interval[0, 𝑧_𝑘(𝑡_𝑘)].

11 Numerical illustrations

This section demonstrates the effectiveness of the algorithm introduced in this chapter and also presents a variety of examples demonstrating two-dimensional scattering in trapping domains. The first example in this section treats the scattering of a unidirectional plane-wave from a kite-shaped geometry. The kite-shaped region is in fact star-shaped, and thus, due to the lack of multiple scattering, the asymptotic

regime is quickly reached. More specifically, the incident field in this experiment is given by the function𝑢^inc(r, 𝑡) =𝑎(𝑡 −k_inc ·r)in the directionk_inc = (1,1) using with signal function

𝑎(𝑡) =sin 𝜋

5

2(𝑡+32) − 40 𝜋 sin

𝜋 40

(𝑡+32)

. (3.5)

The duration of the incident field was taken to be𝑇^inc =3232, so that with a window- width𝐻 =32, there resulted a number of windows𝐾 =100. The frequency-domain was discretized with 491 frequencies in the𝜔-range [0,8], with 191 of those in the range from [0,0.25], so that discretization error was not significant relative to the approximations performed in this section.

The methodology proposed in this section was followed to produce an approximation e𝑢 ≈ Í𝐾

𝑘=1𝑢_𝑘. The function𝑢_𝑘(r, 𝑡) was evaluated at𝑛_𝑠 = 31 sample points𝑧_𝑘(𝑡_𝑛), 1 ≤ 𝑛 ≤ 𝑛_𝑠, equispaced from𝑧_𝑘 =0.25 to𝑧_𝑘 =0.07. Two degree-5 polynomial fits were made, the first for the 16 sample points in [0.16,0.25] and the second for the 16 sample points in [0.07,0.16] (the latter polynomial with the requirement that it is additionally 0 when𝑧_𝑘 =0). The asymptotic behavior for this experiment onset at𝑡_𝑘 =𝑠_𝑘+55, and the polynomial fit was performed on the interval[𝑡_𝑘, 𝑡_𝑘+𝐻]. It should be noted that there were no sample points for𝑧_𝑘 < 0.07, since, under the𝑧_𝑘 change of variables, the value of sample points in the𝑡 variable becomes infinitely large with the result that the size of𝑢may be smaller than the numerical discretiza- tion error from the hybrid methodology presented in Chapter 2—a fact that could otherwise lead to poor interpolation results. In Figure 3.1 the solution𝑢obtained from the full summation Equation (3.3) is compared with the approximation ˜𝑢using the proposed asymptotic fitting methodology. The method produces high-quality approximations that are uniformly-accurate for long times (𝐾 =100 windows).

The numerical test was implemented in MATLAB in a single core on workstation with an Intel Xeon E31230 v5 3.4 GHz CPU. The full evaluation of this long-time simulation required 1/10 seconds per spatial observation point (a figure arrived at by averaging the required time for many spatial observation points). This computation involves evaluate a significant number of Hankel function evaluations, evaluations which can be reused for each window. In order to evaluate the Hankel functions efficiently, certain addition theorems of the Hankel function were utilized, which split the Hankel function into a smooth function and a product of a smooth and of a explicitly-known singular function. In our experiments, this splitting reduced the time to evaluate the required Hankel functions by a factor of seven.

Figure 3.1: Long-time two-dimensional numerical experiment demonstrating the proposed asymptotic-fitting method. Left: Snapshot of representative near-field solution. Right: Error |𝑢− e𝑢| between the true solution 𝑢 and the result of the asymptotic-fitting strategye𝑢. The error is measured at the marked observation point on the left plot,(−2

√ 2,−2

√ 2).

Figure 3.2: Long-time two-dimensional numerical experiment in a trapping region.

Left: Snapshot of representative near-field solution. Right: One of the solution components𝑢_𝑘 at the spatial location marked on the left plot, computed over more than 1000 time units.

Figure 3.3: Scattering in a two-dimensional trapping geometry, with the intensities being that of the total field. The time sequence starts left-to-right on the first row, and then continues left-to-right on the second row.

A second demonstration focuses on a more strongly-trapping geometry, where an incident plane wave field propagating in the direction(−1,0)with signal function

𝑎(𝑡) =sin 𝜋

2.5

2(𝑡+32) − 40 𝜋 sin

𝜋 40

(𝑡+32) ,

impinges on the obstacle pictured in the left-side of Figure 3.2. The observation point is located at(0,0.5), a point in the opening of the cavity. The slow time-decay of the solution is clearly observable, but so also is the eventual slow asymptotic rate of decay.

To conclude this section, we include some further simulations of this trapping geometry in higher-frequency contexts, with Figure 3.3 and Figure 3.4 showing more clearly the trapping nature of this geometry.

Figure 3.4: Higher-frequency scattering in a two-dimensional trapping geometry, with the intensities being that of the total field. The time sequence starts left-to-right on the first row, and then continues left-to-right on the second row.

C h a p t e r 4

TRACKING ACTIVE TIME WINDOWS VIA NOVEL