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Computation of symmetric, time-periodic solutionsof the vortex sheet with surface tensionDavid M. Ambrosea and Jon Wilkeningb,1

aDepartment of Mathematics, Drexel University, Philadelphia, PA 19104; and bDepartment of Mathematics, University of California, Berkeley, CA 94720

Edited by Alexandre J. Chorin, University of California, Berkeley, CA, and approved December 7, 2009 (received for review September 21, 2009)

A numerical method is introduced for the computation of time-

periodic vortex sheets with surface tension separating two immis-

cible, irrotational, two-dimensional ideal fluids of equal density.

The approach is based on minimizing a nonlinear functional of

the initial conditions and supposed period that is positive unless

the solution is periodic, in which case it is zero. An adjoint-based

optimal control technique is used to efficiently compute the gradi-

ent of this functional. Special care is required to handle singular

integrals in the adjoint formulation. Starting with a solution of

the linearized problem about the flat rest state, a family of smooth,

symmetric breathers is found that, at quarter-period time intervals,alternately pass through a flat state of maximal kinetic energy, and

a rest state in which all the energy is stored as potential energy in

the interface. In some cases, the interface overturns before return-ing to the initial, flat configuration. It is found that the bifurcation

diagram describing these solutions contains several disjoint curves

separated by near-bifurcation events.

adjoint method ∣ bifurcation ∣ fluid interface ∣ optimal control ∣

standing waves

Many complex and rich phenomena in nature are controlledby coherent time-periodic or traveling structures. Although

the computation of traveling waves is often straightforward, mostnumerical methods for computing time-periodic solutions weredesigned with ordinary differential equations in mind and aretoo expensive for partial differential equations (PDEs). We de-

velop an adjoint-based optimal control algorithm for solving gen-eral two-point boundary value problems and use it to perform acomputational study of the existence of time-periodic solutions of the vortex sheet with surface tension, which is the interface be-tween two incompressible, irrotational, inviscid, immiscible fluidsshearing past each other. This system poses many technical chal-lenges for the method, and is of considerable interest in mathe-matical fluid mechanics.

We were drawn to several unique features of this problem.First, although the initial value problem is locally well posed(1–5), singularities can form in finite time due to self-intersection(6, 7) or, more speculatively, through the development curvaturesingularities (8, 9); hence, periodic solutions are special in thatthey are global solutions that remain smooth for all time. Second,asymptotic models of interface problems in fluid mechanics are

often integrable; the KdV and Benjamin–Ono equations are twosuch examples. Craig and Worfolk have disproved a conjecture of Dyachenko and Zakharov on the integrability of free surface hy-drodynamics (10). Nevertheless, studies of periodic solutionsshould help illuminate the connection between free surface flowsfor the full Euler equations and integrable model equations ob-tained in various asymptotic limits. Third, there are many inter-esting questions in fluid mechanics regarding ergodicity, recur-rence (11–13), and the role of viscosity in fluid mixing. Alongthese lines, we note that periodicity is beginning to play an im-portant role in the study of turbulence (14–16). Finally, interfaceproblems in fluid mechanics generally suffer from small divisorproblems that require variants of Nash–Moser and Kolmogorov–

Arnold–Moser theory (17, 18) to study time periodicity. By developing such tools, Plotnikov and Toland (19) and Iooss et al.

(20) have proved existence of time-periodic gravity-driven water waves. We aim to learn more about time-periodic interface pro-blems by developing robust numerical methods capable of solvingsuch problems whether or not small divisors are present.

Our numerical method involves two key ideas. First, by adapt-ing adjoint-based optimal control methods (21–24) originally developed in the shape optimization community, we are ableto use quasi-Newton line search algorithms such as the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method (25) to solve two-point boundary value problems rather than the standard methodsof orthogonal collocation (26) or shooting (27). This leads to atremendous reduction in computational cost, especially when ap-proximate Hessian information from the previous solution is usedin the continuation algorithm. The method is a variant of the onedeveloped by the authors in (28, 29) for the Benjamin–Ono equa-tion, but is necessarily more complex as the motion of the vortex sheet with surface tension is described by a coupled system of nonlinear equations rather than a single equation, and involvessingular integrals. Second, to solve the forward and adjoint pro-blems, we use a fourth-order additive Runge–Kutta method (30,31) rather than an implicit–explicit multistep method (32) such asadopted by Hou et al. (6, 7). In either approach, a small-scaledecomposition (developed in ref. 6) is employed in which themost singular terms in the evolution equations are treated impli-citly to remove stiffness, whereas nonlinear terms are treated ex-plicitly. The advantage of the additive Runge–Kutta framework is

that the implicit part of the method is L stable. By contrast, high-order multistep methods lack A stability and must be filtered when used for dispersive problems.

Equations of MotionFollowing refs. 1, 6, 7, we consider two irrotational, idealfluids of equal density separated by a sharp interface, which is acurve ð xðα ; tÞ; yðα ; tÞÞ parametrized by α ∈ ½0; 2π Þ and time. We as-sume the curve is 2π periodic in the horizontal direction, i.e., xðα þ 2π ; tÞ ¼ xðα ; tÞ þ 2π , yðα þ 2π ; tÞ ¼ yðα ; tÞ. The jump inpres-sure across the interface is ½ p ¼ τκ , where τ > 0 is the (constant)coefficient of surface tension and κ is the curvature of the inter-face. We define the arclength element of the curve, ds ¼ σ dα ,and tangent angle, θ , by ðσ cos θ ; σ sin θ Þ ¼ ð xα ; yα Þ. We denotethe tangent and normal vectors to the curve by t

¼ ð xα ∕σ ; y

α

∕σ Þand n ¼ ð− yα ∕σ ; xα ∕σ Þ. We let U denote the normal velocity of

the curve and V the tangential velocity of the curve. We denoteby LðtÞ ¼ 2πσ ðtÞ the length of one period of the curve.

The evolution of θ and σ can be inferred from the evolutionð xt; ytÞ ¼ U n þ V t:

Author contributions: D.M.A. andJ.W. designed research;J.W. performedresearch; D.M.A.

and J.W. contributed new reagents/analytictools; J.W. analyzed data; and D.M.A. and J.W.

wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

1To whom correspondence should be addressed. E-mail: [email protected].

This article contains supporting information online at www.pnas.org/cgi/content/full/

0910830107/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.0910830107 PNAS ∣ February 23, 2010 ∣ vol. 107 ∣ no. 8 ∣ 3361–3366

A P P L I E D

http://www.pnas.org/cgi/content/full/0910830107/DCSupplemental








θ t ¼ U α þ V θ α σ

; σ t ¼ V α − θ α U: [1]

The curve is initially parametrized by arclength, normalized sothat α ∈ ½0; 2π Þ. We choose the tangential velocity to be a non-physical velocity that maintains this normalized arclength para-metrization at all positive times. Thus, we have

σ t ¼ Lt

2π ¼ − P 0½θ α U ; V ¼ ∂−1α P ðθ α U Þ; [2]

where

P 0 f ¼1

2π

Z 2π

0

f ðα Þ dα ; P ¼ id − P 0 [3]

are orthogonal projections onto the mean and onto the space of zero-mean functions, respectively, and ∂−1α is the zero-mean anti-derivative operator.

The normal velocity, U ¼ W · n, is determined by the fluiddynamics via the Birkhoff –Rott integral, W ¼ ðW 1; W 2Þ:

W 1 − iW 2 ¼ 1

2π i PV

Z 2π

0

γ ð β Þ2

cot

zðα Þ − zð β Þ

2

d β : [4]

Here z¼

xþ

iy. The vortex sheet strength, γ , and true vortex sheet strength, ~ γ , are related via γ dα ¼ ~ γ ds or ~ γ ¼ γ ∕σ . The com-plex cotangent comes from summing over periodic images,1

2cot w

2¼ PV ∑ k

1

wþ2π k. The reader could consult, for instance,

the book of Saffman (33) for details on the Birkhoff –Rott inte-gral. The evolution equation for γ is

γ t ¼ ∂α

τ θ α

σ þ ðV − V 1Þγ

σ

; V 1 ¼ W · t; [5]

where V 1 is the average tangential velocity of the fluid across theinterface, ðu þ ivÞÆ ¼ ½∓ðγ ∕2σ Þ þ V 1 þ iU eiθ .

We will need to reconstruct the curve from θ and σ . This isdone by integrating the identity zα ¼ σ eiθ and using zt ¼

ðV

þiU

Þ eiθ to evolve the integration constant:

zðα ; tÞ ¼ ∂−1α P ½σ ðtÞ eiθ ðα ;tÞ þ α þ z0ðtÞ; z0t ¼ P 0ððV þ iU Þ eiθ Þ:

[6]

So we see that to reconstruct the curve, we only need to evolveone point in addition to θ and γ , namely, z0 ¼ P 0 z − π .

If A ≠ 0 and ðσ ðtÞ; θ ðα ; tÞ; γ ðα ; tÞ; z0ðtÞÞ is a solution with surfacetension τ , then ðσ ð AtÞ; θ ðα ; AtÞ; Aγ ðα ; AtÞ; z0ð AtÞÞ is a solution withsurface tension A2τ . Thus, by rescaling time and vortex sheetstrength, we may assume τ ¼ 1. One may also show thatðσ ðtÞ;−θ ðα ; tÞ;−γ ðα ; tÞ; z0ðtÞÞ is another solution.

If at any moment γ ðα ; tÞ and θ ðα ; tÞ are both even functions of α , then the real and imaginary parts of eiθ ðα ;tÞ will be even andthose of ½ zðα ; tÞ − z0ðtÞ will be odd. A change of variables inEq. 4 then shows that W , U , and V 1 are odd functions. Because

V is also odd, γ t and θ t are even, whereas z0t ¼ 0. If, in addition tobeing even, γ and θ change sign upon translation by π , this willalso remain true for all time. In this paper, we look for time-periodic solutions with initial conditions of the form

θ ð·; 0Þ≡ 0; σ ð0Þ ¼ 1; z0ð0Þ ¼ 0;

γ kð0Þ ¼(

0 k even;

cj kj k odd;[7]

where f c k∶ k ¼ 1; 3; 5;…g are real numbers and a hat denotes aFourier coefficient, γ ðα ; tÞ ¼∑ kγ kðtÞ eikα . By the above symmetry arguments, γ k and θ k remain real for all time, and remain zeroif k is even. If at some time T ∕4 the solution with initial conditions[7] evolves to a state in which γ ≡ 0, a time-reversal argument

(with A ¼ −1 above) shows that the solution will evolve back to a flat state at T ∕2 with the sign of γ reversed. The evolutionof γ and θ from T ∕2 to T will be identical to that from 0 to T ∕2,but with opposite signs, ending at the original state.

ResultsThe standard approach to proving existence of time-periodic so-lutions of nonlinear PDE is to build periodicity into the solutionspace and use a Newton iteration (17, 18) to solve a system of lattice equations for the spatial and temporal Fourier modes.Newton’s method converges rapidly enough that “small denomi-nators” can be dealt with via small numerators. Our numericalmethod is based instead on searching for c k and T such that theCauchy problem with initial conditions [7] satisfies γ ð·; T ∕4Þ≡ 0.We define F ðf c kg; T Þ ¼ fγ kðT ∕4Þg k¼1;3;5;… and wish to solve F ¼ 0. If the standard approach were turned into a numericalmethod, it would resemble spectral collocation (26), which is very expensive for PDE. If our approach were used for analytical pur-poses, it would also suffer from small divisor problems.

We begin our search for time-periodic solutions by linearizingthe equations about the flat rest state: z0 ¼ 0, σ ¼ 1, θ t ¼ 1

2 H γ α ,

γ t ¼ τθ αα . Because H ½− sinð kα Þ ¼ cosð kα Þ, the Fourier modessatisfy d

dtθ k ¼ 1

2 kγ k. d

dtγ k ¼ −τ k2θ k. Thus, the solution of the line-

arized problem with initial conditions [7] is θ k

ðt

Þ ¼γ k

ðt

Þ ¼0

when k is even, and

θ kðtÞ ¼ ω k

τ k2cj kj sinðω ktÞ; γ kðtÞ ¼ cj kj cosðω ktÞ [8]

when k is odd. Here ω k ¼ ffiffiffiffiffiffiffi

τ 2 k3

q and T k ¼ 2π

ω k¼ 2

ffiffi2

p π ffiffiffiffiffi

τ k3p are the

angular frequency and period of the kth Fourier mode.If we linearize F about the flat rest state ( c k ¼ 0) with any per-

iod T > 0, we find from Eq. 8 that the Jacobian of F with respectto the c k is an infinite diagonal matrix J (indexed by positive oddintegers) with entries J kk ¼ cosðω kT ∕4Þ, while ∂ F

∂t ¼ 0. A necessary condition for a bifurcation to occur is that J have a nontrivial ker-nel, i.e., there must exist j, k odd and positive so that T ¼ jT k.Fixing T (i.e., k and j), the other entries of J satisfy

A m ≤ j J mmj ≤π

2 A m; A m≔min l odd j m

k3∕2

− l : [9]

Thus, the kernel of J is infinite dimensional (as A kn2 ¼ 0 for odd n) and the range of J is not closed (as A m accumulates at zero,being uniformly distributed (34) over [0, 1]).

Both of these properties prevent a rigorous bifurcation analysisof solutions of F ¼ 0 via the Liapunov–Schmidt reduction (35).Nevertheless, in spite of zero and small divisors, our numericalmethod has no difficulty finding time-periodic solutions. Weuse a solution of the linearized problem (with k ¼ 1) as a startingguess for our optimal control algorithm to find a solution of thenonlinear problem near the flat rest state. We then use numericalcontinuation to increase the amplitude beyond the realm of lineartheory. The continuation algorithm consists of varying one of the

Fourier modes c k0 in [7] of the initial vortex sheet strength, γ 0, andsolving for the other c k and T to minimize the deviation fromtime-periodicity, G ¼ 1

2‖ F ‖2, defined in Eq. 14 below. For each

new value of c k0, we use linear extrapolation from two previously

computed solutions as a starting guess for the remaining c k.In Fig. 1, we show the result of varying c1 from zero (the flat

rest state) to a turning point at −1.08207, and then back up to−0.8321. The solution labeled A on the diagram remains quali-tatively similar to the linearized solutions [8] with k ¼ 1, but high-er frequency Fourier modes become increasingly significant as wecontinue along the bifurcation curve. This diagram contains 1,704time-periodic solutions, each computed down to G ≈ 10−24, withthe number of Fourier modes, M , ranging from 32 to 512. Thesimulations took 4 weeks running simultaneously on fivemachines with a total of 32 cores (running OpenMP, a shared

3362 ∣ www.pnas.org/cgi/doi/10.1073/pnas.0910830107 Ambrose and Wilkening



memory parallel programming language, on each machine). Mostof the running time was devoted to resolving the more compli-cated solutions beyond the turning point in the bifurcation curveand exploring near-bifurcation events (described below). Thepart of the curve connecting the flat rest state to the point labeled

A contains 439 solutions with G ≈ 10−30, but only took 4 h to com-pute on an eight core machine. A few of these solutions were re-computed in double–double precision arithmetic to G ≈ 10−63 tobe sure the algorithm continues to converge when roundoff erroris decreased.

We interpret the turning point as a transition from c1 being thedominant mode to c3 being the dominant mode. In fact, we used c3 as the bifurcation parameter to traverse this region of thecurve. As shown in Fig. 2 (ignoring side branches), c3 decreasesmonotonically through the turning points in T and c1. As we con-tinue along the curve, the solutions develop visibly active second-ary oscillations superimposed on the main carrier wave. In somecases, the vortex sheet briefly overturns before returning to itsinitial flat state.

We noticed small wobbles in some of the plots of c k versus T .By refining the stepsize in the continuation algorithm near each

wobble, we discovered that these curves actually consist of severaldisjoint branches. The 13th Fourier mode c13 of the initial vortex sheet strength gives a particularly nice representation of the“near-bifurcation” events that separate the various branches.

As shown in Fig. 3, these near-bifurcations appear as perturbedpitchforks (35). To our surprise, numerical continuation of theside branches from one of the pitchforks led to reconnections

with the side branches of another pitchfork. One of the branches

appears to be a closed loop.Bifurcation diagrams of still higher Fourier modes reveal ad-

ditional near bifurcations not visible to the first, third, or 13thmode. We illustrate this with the 43rd mode in Fig. 4. A sudden

jump in the curve indicates a transition to a new branch of solu-tions. Following side branches of similar anomalies in lowerFourier modes led to the four branches shown in Figs. 1–3.We stopped following the side-branches of the last two pitchforksin Fig. 3 (and did not follow the new side branches in Fig. 4) as therunning time grew to more than a day per data point (running oneight cores).

We are confident that the disconnection of bifurcationbranches is a true feature of solutions of the PDE rather thana numerical artifact; the curves remain identical (to 9–10 digitsof accuracy) if we cut the mesh size in half. We also emphasize

that the same simulations are shown in all four figures; the addi-tional bifurcations visible in the 43rd mode are a result of lookingat a higher-frequency mode, not a result of running the simula-tions with a smaller mesh size.

We suspect that the disconnection of the bifurcation curves isrelated to resonances and small divisors. In previous studies of nonlinear wave equations (17, 18), it was found that periodicsolutions may not occur in smooth families—their existence couldonly be established for values of a parameter in a Cantor set. Weseem to be observing exactly this phenomenon. The remarkablething is that low-frequency modes are mostly determined by theirinteraction with each other; a sudden jump in a high-frequency mode has little effect. This is why it is possible to compute thesesolutions numerically.

Numerical MethodWe now describe our algorithm for computing time-periodic solutions of the

vortex sheet with surface tension. For the symmetric solutions studied in this

paper, z 0ðt Þ remains zero for all time, so we drop it from the equations in the

interest of brevity. Let q ¼ ðσ ; θ ; γ Þ and define the inner product

Fig. 1. (Left ) Bifurcation from the flat rest state to a family of symmetric breathers, using c 1 ¼ γ 1ð0Þ as the bifurcation parameter. The surface tension τ ¼ 1 is

held fixed. There is a turning point beyond which the first Fourier mode decreases in amplitude while other modes continue to increase. ( Center ) On closer

inspection, the graph consists of at least four distinct branches separated by several near-bifurcation events. (Right ) Time-elapsed snapshots over a quarter-

period of the solutions labeled A and B in the diagrams. Higher frequency oscillations are visibly active in the solution labeled B, which briefly overturns near

t ¼ T ∕4 and 3T ∕4. Movies of these simulations are available as Movies S1 and S2.

Fig. 2. Other Fourier modes can also be used in bifurcation diagrams of this

family of time-periodic solutions. The third mode continues to decrease

through the turning point of the first mode in Fig. 1.

Ambrose and Wilkening PNAS ∣ February 23, 2010 ∣ vol. 107 ∣ no. 8 ∣ 3363

A P P L I E D

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h q1; q2i ¼ σ 1σ 2 þ 1

2π

Z 2π

0

½θ 1ðα Þθ 2ðα Þ þ γ 1ðα Þγ 2ðα Þ dα : [10]

We adapt the small-scale decomposition (SSD) algorithm (6, 7) from the

multistep framework to the additive Runge–Kutta framework and write

the vortex sheet system in the form

qt ¼ f ð qÞ ¼ f 1ð qÞ þ f 2ð qÞ; [11]

where

f 1ð qÞ ¼0

1

σ ð 1

2σ H γ Þα

ðτ

σ

θ α Þα

0@

1A

; f 2ð qÞ ¼− P 0½θ α U

1

σ ½ðU 1Þα þ θ α V

½1

σ ðV − V 1

Þγ

α

0@

1A

:

Here

H γ ðα Þ ¼ 1

π PV

Z ∞

−∞

γ ð β Þα − β

d β ¼ 1

π PV

Z 2π

0

γ ð β Þ2

cot

α − β

2

d β

is the Hilbert transform, which has symbol H k ¼ −i sgnðk Þ. We have desingu-

larized the Birkhoff–Rott integral by writing U ¼ U 1 þ U 2 with

U 1 þ iV 1 ¼ 1

2πσ

Z 2π

0

γ ð β Þ K ðα ; β Þ d β ; U 2 ¼ H γ ðα Þ2σ

;

K ðα ; β Þ ¼ z0ðα Þ2

cot


2

−

1

2cot

α − β

2

: [12]

We have suppressed the dependence of γ and z on time in the notation. The

idea behind the decomposition [11] is to treat the nonlinear operator f 2ðqÞexplicitly and the linear operator f 1ðqÞ, which is the source of stiffness,

implicitly. This is done using two s-stage Butcher arrays (36), one for f 1and another for f 2. In the more general case that f 1 and f 2 depend on time

(e.g., in the adjoint system described below), we define two sets of stage

derivatives and an update step via

ki ¼ f 1

t n þ τ i h; q n þ h∑

j

aij k j þ h∑ j

^ aijℓ j

;

ℓi ¼ f 2

t n þ τ i h; q n þ h∑

j

aij k j þ h∑ j

^ aijℓ j

;

q nþ1 ¼ q n þ h∑ j

b j k j þ h∑ j

^ b jℓ j: [13]

Here h ¼ Δt is the timestep, spatial derivatives and the Hilbert transform are

computed via the fastFourier transform(FFT), and multiplications are donein

physical (as opposed to Fourier) space. The trapezoidal rule is used to eval-

uate the integral in Eq. 12, using K ðα ;α Þ ¼ z αα ðα Þ∕2 z α ðα Þ. We do not simplify

z αα ðα Þ∕2 z α ðα Þ ¼ ði ∕2Þθ α ðα Þ as this identity only holds to Oðh2Þ in internal

Runge–Kutta stages. (The final Runge–Kutta update is nevertheless fourth

order, i.e., Oðh5Þ.) The Butcher array for f 1 is diagonally implicit (aij ¼ 0

for i < j ), whereas that for f 2 is explicit (aij ¼ 0 for i ≤ j ). This allows the stage

derivatives to be solved for in order: k 1 ; ℓ1 ;… ; k s ;ℓ s. In our code, we used the

six-stage fourth-order scheme ARK4(3)6L[2]SA described in ref. 31. If f 2 ¼ 0,

this scheme is stiffly accurate (36), and hence L stable.

Next we definea functionalGðq0 ; T Þ of the initial conditions and supposed

period that is zero if and only if the solution is time periodic. Following pre-

vious work on the Benjamin–

Ono equation (28, 29), we could defineG ¼ 1

2‖qð· ; T Þ − q0‖2, where q solves Eq. 11 with initial condition q0. Instead,

to achieve a factor of four improvement in speed and to emphasize that the

method will work for any two-point boundary value problem (beyond the

computation of time-periodic solutions), we define

Gð q0; T Þ ¼ 1

4π

Z 2π

0

γ ðα ; T Þ2 dα ; [14]

where γ is the third component of q, which satisfies Eq. 11 with initial con-

ditions qð0Þ ¼ q0 to be determined. As in [7], we take q0 of the form σ 0 ¼ 1,

θ 0 ≡ 0, and γ 0 ¼ ∑ðk oddÞc jk jeikx , c k ∈ R. We note that T is now one-quarter of

the period, which is our convention in this section only.

We vary T and the c k in [7] to minimize G using an arbitrary precision C++

version of the limited memory BFGS algorithm (25) we wrote for this project.

BFGS is a quasi-Newton line search algorithm that builds an approximate(inverse) Hessian matrix from the sequence of gradient vectors it encounters

during the course of the line searches. In our continuation algorithm, we

initialize the approximate Hessian with that of the previous minimization

step (rather than the identity matrix), which leads to a tremendous reduction

in the number of iterations required to converge (by factors of 10–20 inmany

cases). We use the limited memory feature of the code for the opposite rea-

son it was originally intended: We store twice as many Hessian updates as

there are columns in the matrix before cyclically overwriting them, which

gives the algorithm more time to achieve superlinear convergence in the

final iterations. The cost of the linear algebra in the BFGS algorithm is dwarf-

ed by the PDE solves required to compute G and ∇G, so there is no benefit to

using fewer Hessian updates. On the other hand, using more than twice as

many columns does not seem to improve convergence rates.

It remains to explain how to compute ∇G, which is needed by the BFGS

algorithm. The T derivative is easily found by evaluating

Fig. 4. The 43rd Fourier mode reveals additional near-bifurcation events not

visible to the first, third, and 13th modes.

Fig. 3. When the 13th Fourier mode of the initial vortex sheet strength is

plotted versus the period, the near-bifurcation events that were almost in-

visible to the first and third modes appear as perturbed pitchforks.




∂G

∂T ¼ 1

2π

Z 2π

0

γ ðα ; T Þγ tðα ; T Þ dα

using the trapezoidal rule. Both quantities γ ð· ; T Þ and γ t ð· ; T Þ are already

known from solving Eq. 11. One way to compute _G ¼ ∂G∕∂c k with k a

positive, odd integer would be to define _q0 ¼ ð0 ; 0 ; eikx þ e−ikx Þ and solve

the variational equation

_ qt ¼ Df ð qð·; tÞÞ_ q [15]

with initial conditions _qð· ;0Þ ¼ _q0 to obtain _γ ðα ; T Þ in

_G ¼ d

dε

ε¼0

Gð q0 þ ε_ q0; T Þ ¼ 1

2π

Z 2π

0

γ ðα ; T Þ_γ ðα ; T Þ dα : [16]

To avoid the expense of solving Eq. 15 repeatedly (for each value of k ), we

solve a single adjoint PDE to find the function δ Gδ q0

ðα Þ ¼ ~ qðα ; T Þ such that

_G ¼ h~ qð·; T Þ; _ q0i ¼ 2 Ref ~ γ kðT Þg: [17]

Here ~ q ¼ ð ~ σ ; ~ θ ; ~ γ Þ are adjoint variables, whereas ~ γ k ðT Þ is the k th Fourier series

coefficient of ~ γ ðα ; T Þ. The function ~ qðα ; sÞ is chosen so that

h~

qð·; T − tÞ;_

qð·; tÞi [18]

is independent of t . When t ¼ T , we put ~ qðα ; 0Þ ¼ ~ q0ðα Þ ¼ ð0 ; 0 ; γ ðα ; T ÞÞ so

that [18] is equal to _G in Eq. 16. When t ¼ 0 in [18], we recover Eq. 17. A

sufficient condition for [18] to remain constant may be obtained by differ-

entiation. This yields the adjoint equation

~ q s ¼ Df ð qð·; T − sÞÞÃ ~ q; [19]

where s ¼ T − t denotes “reversed” time. Like the variational equation,

Eq. 15, the adjoint equation is linear and nonautonomous due to the pre-

sence of the solution qðt Þ in the equation. Note that Eq. 19 only needs to

be solved once to obtain all the derivatives ∂G∕∂c k simultaneously (after one

additional FFT in Eq. 17). Thus, ∇G can be computed in approximately the

same amount of time as G.

To solve the adjoint equation numerically in the additive Runge–Kuttaframework, the values of qð· ; T − sÞ are needed between timesteps (due to

τ i and τ i in Eq. 13), and a small-scale decomposition must be chosen. We

use cubic Hermite interpolation to compute q at these intermediate times,

having stored q and qt at each timestep when Eq. 11 was solved. This is

enough to achieve fourth-order accuracy in the adjoint problem. Our SSD

algorithm is described below.

Due to the presence of singular integrals in Eq. 11, the variational and

adjoint equations are rather complicated. To write down the adjoint equa-

tion, Eq. 19, we must first find formulas forDf ðqÞ _q in Eq. 15. This requires the

intermediate quantities _ z , _U 1 þ i _V 1, _U , and _V to be computed. As always, a

dot indicates a directional derivative withrespect toq in the _q direction. From

Eq. 6, we have

_ z

¼∂−1α Peiθ _σ

þσ ∂−1α Pieiθ _θ ; [20]

where all factors to the right of a projection are multiplied before applying

the projection. Next, from Eq. 12, we obtain

_U 1 þ i _V 1 ¼ −U 1 þ iV 1

σ _σ þ 1

2πσ

Z 2π

0

_γ ð β Þ K ðα ; β Þ d β

þ 1

2πσ

Z 2π

0

γ ð β Þ_ zðα Þ − _ zð β Þ

2cot


2

α

d β :

[21]

The last term is found by writing K ðα ; β Þ in Eq. 12 as an α -derivative and inter-

changing the order of differentiation when the dot is applied. As β → α , the

derivative of the term in brackets approaches z α _ z αα −_ z α z αα 2 z 2α

, so it is not a singular

integral. Next, from U

¼U 1

þU 2, U 2

¼1

2σ H γ , and V

¼∂−1α P

ðθ α U

Þ, we obtain

_U ¼ _U 1 −U 2σ

_σ þ 1

2σ H _γ ; _V ¼ ∂−1α P ð_θ α U þ θ α

_U Þ: [22]

It then follows from Eq. 11 that

Df ð qÞ_ q ¼− P 0½_θ α U þ θ α

_U − θ t

σ _σ þ 1

σ ½ _U α þ _θ α V þ θ α

_V − γ t

σ

_σ

þ1

σ ðτ _θ αα

þ ½_V 2γ

þV 2 _γ

α

Þ

0

B@

1

CA; [23]

where V 2 ¼ V − V 1 and _V 2 ¼ _V − _V 1.

Our final task is to identify the adjoint operator Df ðqÞÃ. Eqs. 20–23 can be

combined into a composition of linear operators, Df ðqÞ ¼ABC , where

_σ _θ

_γ

0@

1A C

↦

_σ _θ

_γ

_ z

0BB@

1CCA B

↦

_σ _θ

_γ _U 1 þ i _V 1

0BB@

1CCA A

↦

_σ t_θ t_γ t

0@

1A: [24]

We then have Df ðqÞÃ ¼ C ÃBÃ AÃ. When computing adjoints, the middle two

spaces in [24] are treated as real inner product spaces with the imaginary

component of the last entry acting as another real dimension, e.g.,

hð_ q1; _ z1Þ; ð_ q2; _ z2Þi ¼ h_ q1; _ q2i þ1

2π

Z 2π 0

Ref_ z1ðα Þ_ z2ðα Þg dα :

Multiplication of _ z by i is interpreted as a rotation by 90° in this real vector

space. From Eq. 20, we obtain

CÃ ¼1 −Re P 0 e

−iθ ∂−1α P id Reiσ e−iθ ∂−1α P

id 0

0@

1A: [25]

Without parentheses, operators and multiplication are always resolved right

to left in our formulas. Similarly, from Eq. 21, we find that

BÃ ¼

1 BÃ41

id 0

id BÃ43

BÃ44

0BBBB@1CCCCA;

BÃ41

w ¼ −σ −1 ReP 0ðU 1 − iV 1Þ w;

BÃ43

w ¼ R 2π 0

Re wð β Þ K ð β ;α Þ2πσ

d β ;

where _U 1 þ i _V 1 ¼ B41 _σ þ B43 _γ þ B44 _ z in Eq. 21. Note that as P 0 is not enclosed

in parentheses in the formula for BÃ41w , we multiply (U 1 − iV 1) by w before

applying P 0 and then taking the real part. Next, we seek BÃ44

such that

1

2π

Z 2π

0

Ref_ zðα Þ BÃ44

wðα Þg dα

¼

Re

4π 2σ Z 2π

0 Z 2π

0

γ

ð β

Þ w

ðα

ÞΔ_ z

2cot

Δ z

2 α d β dα [26]

for all sufficiently smooth test functionsw ðα Þ in L2ð0 ; 2π Þ. Here Δ_ z and Δ z are

shorthand for _ z ðα Þ − _ z ð β Þ and z ðα Þ − z ð β Þ, respectively. As it stands, the singu-

larityin cotðΔ z ∕2Þ as β → α is cancelled byΔ_ z . However, we must separate _ z ðα Þfrom _ z ð β Þ to achieve the desired form on the left-hand side of Eq. 26, which

gives rise to singular integrals. One approach is to write

1

2cot

Δ z

2

¼ 1

zα ðα Þ K ðα ; β Þ þ 1

2cot

α − β

2

with K as in Eq. 12 to convert the singular part of the integral [26] into a

Hilbert transform before separating Δ_ z . Instead, we use the fact thatΔ_ z 2

cotðΔ z 2Þ remains constant if α and β are interchanged. Thus, Eq. 26 may

be written

Ambrose and Wilkening PNAS ∣ February 23, 2010 ∣ vol. 107 ∣ no. 8 ∣ 3365

A P P L I E D



Re

4π 2σ

ZZ −γ ð β Þ w0ðα Þ þ γ ðα Þ w0ð β Þ

2cot

Δ z

2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð⋆Þ

Δ_ z

2d β dα :

We convert this to a principal value integral over the region

S ε ¼ fðα ; β Þ ∈ ½0 ;2π Þ2∶mink ∈ Z jα − β þ 2π k j > εg with ε→ 0, and then use the

fact that (⋆) changes sign when α and β are interchanged to conclude thatΔ_ z 2

may be replaced by _ z ðα Þ. Because γ ðα Þ is real valued, we get the desired

formula

BÃ44

w

ðα

Þ ¼

PV

2πσ Z

2π

0

γ ð β Þ w0ðα Þ þ γ ðα Þ w0ð β Þ

−2

cotΔ z

2 d β :

We evaluate this integral numerically using the trapezoidal rule

ð BÃ44

wÞ k ¼ 1

σ M ∑ j≠ k

−γ j w

0 k þ γ k w

0 j

2cot

z k − z j

2

þ 1

σ M

γ 0 k w0 k z

0 k þ γ k w

00 k z

0 k − γ k w

0 k z

00 k

z0 k2

; [37]

wherea subscriptk indicates evaluation at one of the grid points α k ¼ 2π k ∕M

(0 ≤ k < M ), and primes are α -derivatives computed via the FFT. The j ¼ k term is the ε→ 0 limit of the average of the two values of the integrand

at β ¼ α k Æ ε, weighted by 1∕σ M . The same formulas are obtained if the

integrand is desingularized before applying the trapezoidal rule; hence,

the method is spectrally accurate.

The operator A in [24] may be found by combining Eqs. 22 and 23.

Although tedious, the procedure of forming A and computing the adjoint

AÃ term by term is routine. The result is given in Fig. 5. The terms in boxes

are separated from the rest and treated implicitly in the Runge–Kutta

method; these terms propagate through BÃ and C Ã unaltered. Note that,

although DF ðqÞÃ is linear, a fully implicit approach is impractical as the fulloperator cannot be inverted via the FFT.

ACKNOWLEDGMENTS.This work wassupported in part by theNational Science

Foundation through Grant DMS-0926378 (to D.M.A.) and by the Director,

Office of Science, Computational and Technology Research, US Department

of Energy under Contract DE-AC02-05CH11231 (to J.W.).

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Fig. 5. To compute AÃ ~ q, we apply operators to the components of ~ q from the left and evaluate intermediate operators and multiplications from right to left.

For example, ∂α V acts on~

θ to give ∂α ðV ~

θ Þ while P 0ðθ α ∂−1α P θ α U 2Þ gives P 0½ðθ α Þð∂

−1α ðP ðθ α U 2ÞÞÞð

~

θ Þ. Parentheses only terminate operators enclosed within them, sothe argument of P 0 in the second example includes ~ θ . The terms in boxes are treated implicitly in the additive Runge–Kutta algorithm.


david m. ambrose and jon wilkening- computation of symmetric, time-periodic solutions of the vortex...

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