EVALUATION OF NEAR-SINGULAR INTEGRALS WITH …

EVALUATION OF NEAR-SINGULAR INTEGRALS WITH APPLICATION TO

VORTEX SHEET FLOW

MONIKA NITSCHEUNIVERSITY OF NEW MEXICO, ALBUQUERQUE, NM 87108, USA

Abstract. This paper presents a method to evaluate the near-singular line integrals that describe thesolution of elliptic boundary value problems in planar and axisymmetric geometries. The integrals arenear-singular for target points not on, but near the boundary, and standard quadratures loose accuracyas the distance d to the boundary decreases. The method is based on Taylor series approximations of theintegrands that capture the near-singular behaviour and can be integrated in closed form. It amountsto applying the trapezoid rule using meshsize h, and adding a correction for each of the basis functionsin the Taylor series. The corrections are computed at a cost of O(1) per target point. The number ofcorrections required for a desired order of accuracy is determined by error bounds obtained for each basisfunction. Two explicit versions of the method of order O(dh, h4) and O(d2h, h4) are listed. The resultingintegration error practically vanishes as d → 0 and attains maximum values of order O(h2) and O(h3),respectively, at a distance d ∼ h. The method is applied to compute planar potential flow past a plateand past two cylinders. The observed results are consistent with the analytically predicted convergencerates.

1. Introduction

Elliptic boundary value problems can be formulated as boundary integral equations in terms of aproblem-dependent Green’s function and a density function. For example, potential flow past objects isdescribed by a vortex sheet bound to the surface of the object whose density, the sheet strength, is suchthat the normal velocity vanishes on the walls. Rigid or flexible vesicle motion in Stokes flow is givenby a sheet of Stokeslets whose density, the normal stress force, is also such that boundary conditionsare satisfied. Numerical methods based on solving the boundary integral equations (BIE) are efficient,as they reduce the problem to a lower-dimensional one, accurate, as they track the boundary with highresolution, and flexible, as multiple domains and moving boundaries are easily addressed.

For domains in R3 this formulation yields boundary surface integrals. When planar or axial symmetryis present the problem reduces to a domain in R2 with boundary line integrals. This paper concernsthe line integrals that arise in planar or axisymmetric geometries. The BIE methods require evaluatingthe line integrals at target points on the boundary, either to solve for the density or to obtain theself-induced velocity, and also at points away from the boundary, for example to compute the motionof particles or other interfaces in the fluid. For target points on the boundary the integrals are singularand can be evaluated accurately using quadrature rules such as given by Sidi and Israeli [15]. Fortarget points far from the boundary the integrals are regular and can be evaluated accurately using, forexample, variants of the trapezoid rule. However, for target points not on the boundary but near it,the integrands are near-singular and standard quadrature rules loose accuracy. As an example of thedifficulty this causes, consider multiple drops close to each other moving in Stokes flow. Inaccuraciesnear the drop interfaces can lead to drop boundary crossings, after which the solution breaks down.

The focus of this paper is the accurate evaluation of the near-singular integrals. This topic has beenthe subject of much recent research, motivated by solutions to the Laplace, Helmholtz, and Stokesequations. Several accurate approaches have been proposed. Ying, Biros and Zorin [19] obtain thesolution near the boundary by interpolating values on the boundary and values far from it. Thismethod is applied by Quaife and Biros [13] to compute 2D vesicle motion in Stokes flow. Several worksaddress planar problems by framing them in terms of complex variables: Helsing and Ojala [7] proposean interpolatory panel method for Laplace’s equation based on integrating a basis of complex functions;Ojala and Tornberg [11] adapted that method to multiphase Stokes flow; Barnett, Wu and Veerapaneni

1

2 MONIKA NITSCHE UNIVERSITY OF NEW MEXICO, ALBUQUERQUE, NM 87108, USA

C

x0

x( )α

d

Figure 1. Sketch showing curve C given by x(α) with target point x0 at distance d from C.

[1] present a method based on a baricentric formula for Cauchy integrals. A method in which the densityis interpolated and smoothed has been recently proposed by Perez-Arancibia, Faria, and Turc [12] andapplied in both 2D and 3D. A different approach by Beale and Lai [3] consists of regularizing the near-singular integrands, integrating the smoother functions accurately, and adding corrections obtained byasymptotic approximations of the regularization error. This method has been extended to 3D surfaceintegrals (see Tlupova and Beale [17, 18] and Beale, Ying and Wilson [4]), and has been applied byCortez to compute regularized Stokes flow [6]. In a further line of research Carvalho, Khatri and Kim [5]derived matched asymptotic expansions for target points near the boundaries. Another recent approachis the method of quadrature by expansion (QBX) introduced by Klockner et al [10], based on the workof Barnett [2]. This method consists of finding high order kernel expansions about points in the interiorof the domain, in the region resolved accurately by standard quadratures, and applying these expansionsto resolve the flow near the boundary. The QBX method has been shown to give accurate results inboth 2D and 3D applications, although it is nontrivial to implement (eg, see Klinteberg and Tornberg[8, 9] and Siegel and Tornberg [16]). Rahimian, Barnett and Zorin [14] have recently extended themethod to a kernel-independent approach.

Here we propose a simple method to evaluate near-singular line integrals of the following form,

(1)

∫ b

a

F2(x(α),x0)

|x(α)− x0|4ω(α) dα ,

∫ b

a

F1(x(α),x0)

|x(α)− x0|2ω(α) dα ,

∫ b

alog |x(α)− x0|2 ω(α) dα.

where x(α), α ∈ [a, b] is a smooth parametrization of a curve C in the plane, F1,2(x,x0) and ω(α) aresmooth, and the target point x0 is not on the curve C, but at a distance d, as illustrated in figure1. These are the types of integrals that arise in planar vortex sheet flow and axisymmetric Stokesflow. They are near-singular if d is small. It is important to note that since these integrals stem fromboundary integral formulations, they are integrable in at least the principal value sense when x0 ∈ C.Thus the numerators F1 and F2 encountered in applications have at least a simple and a triple rootwhen x = x0, respectively.

The method proposed here consists of approximating each integrand by a function that captures thenear singular behaviour and can be integrated analytically. The approximating function is obtained byTaylor series expansions about a point α = αp on the boundary, as follows: if a given target point x0 isfound to be too close to the boundary, we find the orthogonal projection xp = x(αp) of x0 onto C andexpand the integrand about αp in terms of elementary basis functions. The implementation consistsof computing a correction for each of the basis functions and adding it to a basic 4th order trapezoidapproximation of the integral.

The corrections are computed at a cost of O(1) per target point. The number of corrections neededdepends on the desired order of accuracy. For example, the middle integral in (1) requires 5 correctionsfor a second order method and 10 corrections for a third order method. The method relies solely onknowing the expressions for F and ω, and is therefore not restricted to planar geometries. In principle,it is also applicable to 3D surface integrals. Here we will apply it to two sample planar vortex sheetflows. We note that in planar vortex sheet flow, the integrals are particularly simple in form, presentinga simple framework to introduce the method. The approach taken is however quite general and hasalready been extended to axisymmetric Stokes flow, although the details in that case are more complexand will be presented elsewhere.

The method is conceptually simple, simple to implement, and accurate near the boundary. In par-ticular, the resulting approximation error is shown to basically vanish as the target point approachesthe boundary. For example, for the second and third order methods referred to above, the maximumerror is shown to be O(hd, h4) and O(hd2, h4), where d is, as before, the distance from the target point

EVALUATION OF NEAR-SINGULAR INTEGRALS WITH APPLICATION TO VORTEX SHEET FLOW 3

to the boundary and h is the meshsize used for the underlying trapezoid rule. Here we choose a 4thorder variant of the trapezoid rule which is the source of the O(h4) term. These results imply almostvanishing error as d→ 0, for fixed h, and maximal errors of order O(h2) and O(h3) when d ∼ h.

The paper is organized as follows. Section 2 describes the method, motivated by a simple example.Section 3 applies the method to compute planar potential flow past a plate and past two cylinders.These two examples present a different set of difficulties, as will be discussed. Section 4 presents thederivation of the order of convergence. The work is summarized in section 5.

2. Numerical Method

This section describes the proposed method to evaluate integrals of the form (1) for small values of d.We assume that x(α) is a smooth parametrization of a smooth curve C, and F, ω are smooth functionsof α. In section 2.1, we first present a simple example that illustrates the difficulty in evaluatingthe integrals and motivates the method. Sections 2.2-2.4 describe the method. Section 2.5 states theconvergence results that are proved later, in section 4.

2.1. Motivating example. Consider a flat curve C given by x(α) = (α, 0), uniformly discretized bypoints αk = kh, and a target point x0 positioned near the origin with coordinates (αp, d), as illustratedin figure 2. The projection αp is at a distance ah from the nearest gridpoint. Now consider the errormade by the trapezoid rule in evaluating the integral

(2)

∫I

F1(x(α),x0)

|x(α)− x0|2dα

over a small interval I = [−h, h], for simple numerators F1 = 1 or F1 = α− αp. In this simple example|x(α)− x0|2 = d2 + (α− αp)2, see figure 2, and the integrand reduces to

(3) f1(α) =1

d2 + (α− αp)2or f2(α) =

α− αpd2 + (α− αp)2

.

Figure 3(a) plots f1(α) for a generic value of d � h, with αp = 0. The maximum value fmax = 1/d2

and a characteristic width, here the width 4d when f1 = fmax/5, are indicated. These characteristiclength scales indicate the behaviour of the graph as d → 0. The shaded area represents the trapezoidrule error

(4) E[f ] =

∫f − T [f ]

over the interval I. The minus signs indicate that the contribution of each area is negative. It is clearfrom the figure that in magnitude, the error is the area of two almost triangular regions and that, ifd� h,

E[f1] ≈ −h/d2 .The absolute error thus tends to infinity as d→ 0, for any fixed meshsize h.

Figure 2. Motivating example. Flat curve C given by x(α) = α, discretized by equally spacedpoints αk = kh. The point αp is the orthogonal projection onto the curve of the point x0 atdistance d. The distance from αp to the closest gridpoint is measured by ah.


Figure 3. Graphs of (a) f1 with αp = 0, (b) f2 with αp = 0 and (c), f2 with αp > 0, for d� h.The shaded area represents the error E[f ] on the interval [−h, h]. The sign of the the error oneach piece is denoted by “-” and “+”.

For f2 the situation is slightly different. The function f2 is odd about αp and one may think that thetrapezoid error is zero by cancellation. This is so in the case αp = 0 shown in figure 3(b),

E[f2] = 0 if αp = 0 .

However, it is not true if αp 6= 0. In the case αp = d, shown in figure 3(c), the function takes on itsmaximal magnitude of 1/(2d) at a gridpoint, namely the middle gridpoint. The figure shows that as aresult, the contributions of the shaded areas to the error are both positive and there is no cancellation. Itis furthermore easy to see that the shaded area to the right of αp and above the axis approximately equalsthe white area to the left between the axis and the function. Thus the total shaded area approximatelyequals that of a triangle with height 1/(2d) and width 2h. We deduce that for d � h, the trapezoiderror

E[f2] ≈ h/(2d) if αp = d ,

which also approaches infinity as d → 0. Thus, in the case of f2 the error oscillates greatly, between0 and h/(2d), as the target point x0 moves at a constant distance d parallel to the curve C. Thisillustrates the importance of the parameter ah shown in figure 2.

This simple scenario illustrates the difficulty in accurately evaluating the near-singular integralsusing standard quadratures. However, it also motivates the method proposed next: the key idea isthat functions of type f1, f2 are leading order approximations to the integrands and can be integratedexactly.

2.2. The Method in a Nutshell. We denote the generic integrands in (1), for given x(α) and x0, by

(5) G2(α) =F2(x(α),x0)

|x(α)− x0|4ω(α) , G1(α) =

F1(x(α),x0)

|x(α)− x0|2ω(α) , G0(α) = log |x(α)− x0|2ω(α).

The distance from x0 to the curve is d, and h is the meshsize of a uniform discretization {αk}nk=0 of[a, b]. The main idea is to approximate the near-singular functions G(α) to leading order for d� h by a

Figure 4. Projection xp = x(αp) of x0 onto curve given by x(α), which is discretizedby points x(αk). The point x(αk0) is the gridpoint closest to xp.


function H(α) that captures the near-singularity and can be integrated exactly. We then approximatethe integral of G as follows: ∫

G =

∫(G−H+H)(6a)

≈ T4[G−H] +

∫H(6b)

= T4[G] +( ∫H− T4[H]

)(6c)

= T4[G] + E4[H] .(6d)

In (6a), we add and subtract the approximation fromG. In (6b), we approximate the smoother differenceG−H by a fourth order generalized trapezoid rule T4, while integrating H exactly. In (6c), we rearrangeterms using the linearity of T4. Equation (6d) summarizes the method as implemented: it consists ofadding the correction E4[H] =

∫H− T4[H] to the trapezoid approximation of the uncorrected integral.

For T4 we use

(7) T4[f ] = h

n∑k=0

′f(αk)−h2

12[f ′(b)− f ′(a)] ,

where the prime on the summation denotes that the first and last term in the sum are weighted by 1/2.This basic approach follows that of earlier work by the author and coworkers in [?, ?, ?].

2.3. The approximation H. We begin by finding the orthogonal projection xp = x(αp) of x0 onto thecurve C (see figure 4). This step determines d = |x0−xp| and the basepoint αp. The approximation Hconsists of a Taylor approximation of G(α) about αp. It is obtained by expanding both the numeratorand the denominator ofG using as many terms as needed for the desired accuracy, applying the geometricseries, and truncating the result. The number of required terms in the two expansions is deduced fromthe convergence results provided in section 4. This section §2.3 presents a method of order O(hd)+O(h4),in short, O(hd, h4). It requires a 4th order expansion of the denominator,

|x(α)− x0|2 =[ (xp − x0) + xp(α− αp) +1

2xp(α− αp)2 +

1

6

...xp(α− αp)3 +O((α− αp)4 ]·

[ (xp − x0) + xp(α− αp) +1

2xp(α− αp)2 +

1

6

...xp(α− αp)3 +O((α− αp)4 ]

=d2 + c2(α− αp)2 + e(α− αp)3 +O((α− αp)4) ,

(8)

where c2 = |xp|2 + xp · (xp − x0) and e = 13(xp − x0) ·

...xp + xp · xp, and xp = x(αp), xp = x(αp),...

xp =...x(αp). We note that in the cases considered below in this paper, of either a flat boundary C,

or a curved boundary uniformly discretized in arclength, c2 is positive. In particular, if x0 lies on the“outside” of a curve, c2 is always positive, and if it lies “inside”, c2 > 0 when d is less than the radiusRosc of the osculating circle at xp. Here the inside is defined to be the side of the osculating circle. Forfurther detail, see Appendix A. The results below assume that c2 is strictly bounded away from zero,which holds if d is strictly less than Rosc.

The number of terms required for the numerator depends on the order of the near-singularity of G.For example, for G1 we use,

(9) F1(x,x0)ω(α) = c0 + c1(α− αp) + c2(α− αp)2 +O((α− αp)3)

where, since F1 must have a simple root at x = x0 as remarked earlier, the constant term is proportionalto d, hence c0 = c0d. Here the tilde denotes a constant of order O(1). This observation will be importantfor the convergence results in section 4. The expansions (8,9) are then used to expand G1 about αp and


approximate it as follows, where for convenience we let α = α− αp,

G1(α) =F (x(α),x0)

|x(α)− x0|2ω(α) =

c0 + c1α+ c2α2 +O(α3)

d2 + c2α2 + eα3 +O(α4)(10a)

=c0 + c1α+ c2α

2 +O(α3)

d2 + c2α2

[1− eα3 +O(α4)

d2 + c2α2

](10b)

≈ H1(α) =c0 + c1α+ c2α

2

d2 + c2α2− e c0α

3 + c1α4

(d2 + c2α2)2.(10c)

Note that the geometric series expansion applied in step (10b) converges for sufficiently small α since,even though α3 is not necessarily smaller than d2, α3 is certainly smaller than α2. The approximationH1

given in (10c) is obtained by truncating the expansion of G1. Note that all terms in H1 can be integratedanalytically in closed form, and thus the correction E4[H] can be readily computed. The approximationrequires only second derivatives of xp(α) and ω(α) at xp to obtain the coefficients c0, . . . , c2.

The function G2 is approximated as follows,

G2(α) =F (x(α),x0)

|x(α)− x0|4ω(α) =

c0 + c1α+ c2α2 + c3α

3 + c4α4 +O(α5)

(d2 + c2α2 + eα3 +O(α4))2(11a)

=c0 + c1α+ c2α

2 + c3α3 + c4α

4 +O(α5)

(d2 + c2α2)2

[1− 2

eα3 +O(α4)

d2 + c2α2+

O(α6)

(d2 + c2α2)2

](11b)

≈ H2(α) =c0 + c1α+ c2α

2 + c3α3 + c4α

4

(d2 + c2α2)2− 2e

c0α3 + c1α

4 + c2α5 + c3α

6

(d2 + c2α2)3(11c)

where, in view of the triple root of F2 at x = x0, c0 = c0d3, c1 = c1d

2, c2 = c2d. We note that as aresult, the approximation of G2 also requires only second derivatives of x(α) to obtain the coefficientsc0, . . . , c4.For G0 we use,

G0(α) = ω(α) log |x(α)− x0|2 =[c0 + c1α+ c2α

2 +O(α3)]

log(d2 + c2α2 +O(α3))(12a)

=[c0 + c1α+ c2α

2 +O(α3)][

log(d2 + c2α2) +O(α3)

d2 + c2α2

](12b)

≈ H0(α) = (c0 + c1α+ c2α2) log(d2 + c2α2)(12c)

For each G, the approximations H given in equations (10c)-(12c) are a sum of basis functions of theform

(13) H0k = αk log(d2 + c2α2) , Hjk =αk

(d2 + c2α2)j, j ≥ 1

The implementation given next consists of computing E[H] by first computing E[Hjk] for each requiredbasis function. The analysis presented in section 4 is also based on studying each basis function first.

2.4. Implementation: Specific steps. To implement the method given a target point x0, we takethe following steps:

(1) Compute T4[G].(2) If x0 is too close to C then

(a) find αp, d,xp, xp, xp,...xp to obtain c, e

(b) find the required coefficients c0, c1, c2 (for G1, G0) or c0, c1, c2, c3, c4 (for G2)(c) find the closest gridpoint αk0 to αp(d) compute E4[Hjk], as needed, where Hjk are as in (13).

Here, it is only necessary to evaluate the errors over an interval centered at αk0 ,

(14) I = [αk0−nwin, αk0+nwin

] .


Most results below use nwin = 10 although larger values may be needed, see note (iv).

(e) correct

∫G ≈ T4[G] +

∑c′jkE4[Hjk] where c′jk are the appropriate coefficients in the

expansions (10c,11c,12c). The sum consists of 3, 5 and 9 terms for G0, G1 and G2

respectively.

Several notes are in order:(i) Definition of “too close”: In the calculations in section 3 below, the correction is applied whend < 4∆s, where ∆s is the spacing between the two gridpoints closest to x0. This criterion is consistentwith the criterion used by Klockner et al [10], who considered the “high-accuracy region” in whichstandard quadratures give good results and no correction is necessary to be at a distance of 5h ∼ 5∆s.This is the distance at which the base points of the kernel expansions in [10] are placed.(ii) In step 2a: The coefficients c, e depend on up to third derivatives of the curve x(α) at αp, exceptfor circles discretized by equally spaced points, for which e = 0.(iii) In step 2b: The coefficients ck depend on up to second derivatives of x(α) and ω(α) at αp.(iv) In step 2d: small values of nwin can cause the errors for large d to dominate those for moderatelysmall d. This effect was noticeable for the O(h3) results shown in section 3.1.2 below. To reduce theerrors for large d, those results were obtained with nwin = 40 − 80. A correction for large d couldpotentially be added to be able to use smaller windows.(v) In step 2a: if x0 does not have an orthogonal projection onto C (for example, if it lies to the leftor right of the flat plate considered below) yet is close enough to require correction, we use the closestpoint on C as xp. This does not introduce a linear term into (8) since we always impose xp = 0 at tips.Furthermore, c2 > 0 in that case as well.(vi) In step 2d, for nonperiodic problems only: To accomodate the case when k0 − nwin < 0 we replacethat index by max(k0 − nwin, 0), and similarly k0 + nwin is replaced by min(k0 + nwin, n). This caseoccurs in the flat plate example below when αp is too close to the plate edges.

2.5. Convergence as d → 0. In section 4 we prove that the method presented above converges, andthat for points near the boundary, the resulting integration error is O(hd, h4). Thus it follows that (i)for d ≤ h the maximal errors near the wall are of order O(h2), and (ii) the error vanishes uniformly asd → 0, up to a small term of size O(h4) which is independent of d, a. Before presenting the analyticalresults, section 3 presents numerical results in simple applications. As noted, the order of accuracy ofthe method can be improved using more terms in the Taylor series approximations. To illustrate, wealso show numerical results using an O(h3) method that is derived later, in section 4.

3. Vortex sheet flow

In this section we evaluate the performance of the method in simple test cases, such as illustratedin figure 5: computing planar potential flow past objects (figures 5a,b,c) or vortex sheet separation atthe edge of a plate in a background flow (figure 5d). In each case, the flow is induced by a vortexsheet bound to the object, shown in green, whose sheet strength is such that the normal velocity on theboundary is zero. The velocity and streamfunction induced by the bound sheet at a point x0 not on it

Figure 5. (a-c) Potential flow past plate and cylinders. (d) Vortex sheet separation atedges of inclined plate.


are given by

u(x0) =1

2π

∫ b

a

k× (x− x0)

|x− x0|2Γ′(α) dα+ U∞ ,(15a)

ψ(x0) =1

4π

∫ b

alog |x− x0|2 Γ′(α) dα+ ψ∞ ,(15b)

where x = x(α), α ∈ [a, b], is a parametrization of the sheet, the density Γ′(α) is the scaled sheetstrength, U∞ is the driving external flow and ψ∞ the corresponding streamfunction. When x0 is nearthe boundary, these integrals are near-singular of the type discussed above. Note that in (15a), thesingle root of the numerator F1 at x = x0 is clearly evident. To compute the integrals one must firstdetermine the sheet strength Γ′(α) at the gridpoints. It is given by a Fredholm integral equation of thefirst kind, u(x0) · n = 0 if x0 ∈ C. Here, we compute it numerically, following the approach taken forexample in [?]: the sheet x(α) is discretized by n + 1 uniformly spaced points αk, k = 0, . . . , n. Theno-through flow is imposed at the midpoints αmk = (αk−1+αk)/2, k = 1, . . . , n, and the total circulationis prescribed. These n+ 1 conditions determine Γ′(αk) for all k. The integrands are then known at allgridpoints and the second or fourth order trapezoid approximation can be applied. We note that thesecond order trapezoid rule is equivalent to the point vortex approximation of the sheet.

For brevity, here we show detailed results for two of these examples that present a different set ofdifficulties. The flat plate shown in figure 5(a) consists of a finite sheet with unbounded velocities nearthe two plate edges, and thus presents a test for the method in non-periodic domains, away from aswell as near endpoints and singular points. An explicit expression for the potential flow is known andwill be used for comparison. The two-cylinder problem shown in figure 5(c) consists of two smoothperiodic vortex sheets. Here, the main difficulty lies in the numerical solution for the sheet strengthif the distance between the cylinders is small. The sheet strength on one cylinder, for example at thepoint x0 shown in the figure, depends on the velocity induced by the other cylinder. If the distance issmall, the velocity on the nearby cylinder is given by a near-singular integral. Inaccurate integrationyields highly inaccurate sheet strength. We investigate the sheet strength and the resulting streamlineswith and without the corrections proposed above. The two-cylinder problem is a simple planar versionof the two-sphere problem considered in Siegel and Tornberg [16]. While an exact expression of thesolution may be found, for example using the methods of Crowdy and Marshall [?], our emphasis hereis to determine the accuracy that can be attained numerically.

3.1. Potential flow past a flat plate. The plate is positioned on the x-axis, centered at the originwith length 2, and the upward far field flow is U∞ = 〈0, 1〉. The plate is parametrized by x(α) =〈− cosα, 0〉, α ∈ [0, π], and discretized by

(16) x(αk) , αk = kh , h = π/n , k = 0, . . . , n.

xo

xp

(a)

-1 -0.5 0 0.5 1

x

-1

-0.5

0

0.5

1

y

-2

-1

0

1

2

'()

0 /4 /2 3 /4

(b)

n=50

100

200

400

800

Figure 6. (a) Plate discretized by points, red line positioned at a generic x0 betweenpoints, here x0 = −0.753. (b) Circulation computed using n = 50, 100, 200, 400, 800points, as indicated.


-2 -1 0 1 2

u

-0.05

0

0.05

y

(a)

n=50

100

200

400

800

0 1 2 3

v

-0.05

0

0.05

y

(b)

-2 -1 0 1 2

u

-0.05

0

0.05

y

(d)

n=50

100

200

400

800

0 0.05 0.1 0.15 0.2 0.25 0.3

v

-0.05

0

0.05

y

(e)

Figure 7. (a) Plate discretized by points, red line positioned a generic x0, (b,c) hor-izontal and vertical velocity along red line vs y, computed with trapezoid rule, (d,e)corrected velocity, using n = 50, 100, 200, 400, 800 points, as indicated.

As a result the points are clustered near the edges. Figure 6(a) shows the position of the grid pointsfor n = 50. Figure 6(b) shows the sheet strength on these grid points computed as described above, forn = 50, 100, 200, 400, 800. The different curves are indistinguishable and differ from the exact solutionΓ′(αk) = −2 cosαk by 10−13 − 10−11, for all n, with increasing error as n increases due to increasingcondition number. Thus the approach taken to obtain Γ′(αk) gives sufficiently accurate results and isnot responsible for the errors observed below.

3.1.1. Results of O(h2) corrected quadrature. Figure 6(a) also shows a red line crossing the plate ver-tically at a generic position that avoids the gridpoints. Figures 7(a,b) plot the horizontal and verticalvelocity components u and v along this red line, vs y, obtained by evaluating (15a) using the uncor-rected trapezoid rule T4 with n = 50, 100, 200, 400, 800. In the exact solution, the horizontal velocityu is discontinuous across y = 0, negative below and positive above. The vertical velocity is symmetricacross y, positive on both sides but decreasing to zero as the plate is approached. Figures (a,b) showthat the trapezoid rule recovers this behaviour away from the plate, but the errors near the plate arelarge. Figure (c) plots the magnitude of the error as a function of the distance d from the plate. It showsthat as n increases, the errors are large in an interval of shrinking width, but the maximum error is andremains large for all n. Figures (d,e) show the computed solution after adding the corrections E4[H]described in section 2. Results are shown for all n, yet they are indistinguishable to the eye. Figure (f)plots the errors obtained by comparison to the known exact solution. It shows that corrections havereduced the errors in figure (c) by a factor bigger than 104, already with n = 50. The maximum errorsappear to decrease by a factor close to 4 each time n is doubled, consistent with the assertion abovethat the method is O(h2). Also significant is that, while the maximal error is O(h2) at larger values ofd, the error decreases to almost zero as d→ 0. This is consistent with the assertion that the method isO(hd, h4). The convergence of the method in this sample test case is revisited in more detail in section3.1.2 below.

Figure 8 plots the streamlines past the plate computed by integrating (15b) with the trapezoid rule(figure a) and after adding the corrections (figure b). The uncorrected results clearly show the localrotating fluid motion around the individual vortices discretizing the plate. The correction reproducesparallel flow at the plate walls. This is consistent with the observation in figure 7(f) that the correctedapproximation error almost vanishes as d→ 0.


Figure 8. Streamlines computed with n = 50 and (a) trapezoid rule, (b) corrected quadrature.

While the inaccuracies of the trapezoid approximation occur in a thin region near the plate, the effecton particle motion in the flow is significant. To illustrate, figures 9 and 10 present the evolution of acloud of particles in the computed potential flow, using both the velocity computed by the uncorrectedtrapezoid rule, and the corrected velocity. The particle position is evolved using the 4th order RungeKutta method. The particles are initially distributed in a circular region, with random phase, positionedat distance 0.5 below the plate. Figure 9 shows the computed position at a sequence of times 1.3 ≤ t ≤ 5.Each frame shows the trapezoid results in the left column, and the corrected results in the right column.The initially circular cloud flattens as it approaches the plate. At time t = 2.75, oscillations are visiblein the left column that are not present in the right. At times t = 3.3, 3.9, the particles in the left columnare seen to cross the plate. This is consistent with the streamlines of the point vortex flow shown in

-0.1

0

(a) t=1.30 t=1.30

-0.1

0

0.1 (d) t=3.30 t=3.30

-0.1

0

0.1(b) t=2.00 t=2.00

0

0.1

(e) t=3.90 t=3.90

-0.1

0

0.1 (c) t=2.75 t=2.75

-1 0 1

0

0.1

(f) t=5.00

-1 0 1

t=5.00

Figure 9. Position of cloud of points initially at distance 0.5 below the plate, at theindicated times. Each subplot shows results using Trapezoid rule at left, corrected quad-rature at right.


-0.02

0

0.02

(g) t=3.30 t=3.30

-0.02

0

0.02

(i) t=3.90 t=3.90

-0.02

0

0.02

(h) t=3.50 t=3.50

-0.02

0

0.02

(j) t=5.00 t=5.00

Figure 10. Closeup of solutions at indicated times, with Trapezoid rule at left, correctedquadrature at right.

figure 8a. There is no apparent particle crossing through the plate in the corrected results in the rightcolumn.

Figure 10 shows a closeup of results for 3.3 ≤ t ≤ 5. It clearly shows the particles moving through thewall in the left column, and clearly moving around the plate in the right. We see again that, since thecorrected error vanishes as d→ 0, particles are prevented from moving through the wall. Three moviesshowing the particle motion at three different scales are available for view as supplemental materials ofthis paper.

3.1.2. Observed rate of convergence, O(h2) and O(h3) methods. Figure 7 plotted the error in the com-puted velocity along a line. We now discuss the error in the whole domain. Figure 11 shows the relativemagnitude of the error in the velocity ||u − uex||2/||uex||2, sampled on a fine rectangular grid aroundthe plate, computed with both the uncorrected trapezoid method in the left column (figures a,d) andthe corrected quadrature presented in section 2 in the middle column (figures b,e), using n = 50 pointson the plate in the top row (figures a-c) and n = 800 in the bottom row (figures d-f). Here u and uexare the computed and the exact velocity at a gridpoint x and the norm used is the Euclidean 2-norm.

Above, we asserted that the method proposed in section 2 is O(h2), and that it can be made of anyhigher desired order by including more terms in the Taylor series. To illustrate, figures (c,f) show theerror using the next higher order O(h3) method. This method is derived and presented later, togetherwith the analytic derivation of the order of convergence, in section 4.

The errors in the point vortex approximation in figures (a,d) diverge to infinity as the point vorticesare aproached, when both d and ah → 0. The gridpoints chosen in this figure avoid the plate, andare positioned at distance d ≥ 0.0005h for both n = 50 and n = 800. Thus the maximal errors of thetrapezoid approximation are large but bounded on the grid. The errors are shown on a logarithmiccolor scale from 10−12 to 102, with all errors < 10−12 shown as dark blue, and all errors > 102 shownas bright yellow. Figure (a) clearly shows the large errors near each of the point vortices. The errorsdecrease as the distance to the plate increases. The errors are approximately constant along ellipticallevel curves. In view of the chosen point distribution, the set of points x whose distance to the plateis a constant multiple of the distance ∆s(x) between the two nearest gridpoints on the plate, forms anellipse. Thus the error is proportional to ∆s(x). Consistent with these remarks, note that the verticalscale for n = 800 is a factor of 16 smaller than the vertical scale for n = 50, that is, proportional to∆s(x).


Figure 11. Relative error ||u − uex||2/||uex||2 on a fine grid, computed with (a,d) originaltrapezoid rule, (b,e) O(dh) method, (c,f) O(d2h) method. The top row (a,b,c) shows results forn = 50, the bottom row (d,e,f) for n = 800. The errors are plotted using a logarithmic colorscale, where all errors bigger than 102 and less than 10−12 are incorporated in the top yellow andbottom blue colors.

The correction is applied to points x that are within 4∆s(x). Per the previous discussion, thisrepresents an elliptical region around the plate. All corrected error plots (b,c,e,f) clearly show theelliptical region in which the correction has been applied. Near the plate the corrected errors aresmall, as shown by the dark blue center line. At moderate distances the errors have a bimodal shapewith increasing d, as seen earlier in figure 7(f). At large distances, d ≈ 4∆s, the errors in the O(h2)results in (b,e) are larger than the uncorrected results. This is not the case in the O(h3) results withn = 800 shown in figure (f). There, the corrected results are more accurate than the uncorrected resultseverywhere.

Finally, note that the relative error near the plate edges is somewhat larger than away from the edges.The relative error near the origin is also larger, as visible specially in (f), since the actual velocitiesthere are small.

The maximal errors

(17) maxx∈D

[min

( ||u− uex||2||uex||2

, ||u− uex||2)]

for the three methods shown in figure 11 are plotted in figure 12, versus 1/n = h/π. Note that thefigures plot the maximum of the minimum of the absolute and the relative error to avoid large relativeerrors when the velocity is small. Figures 12(a,b,c) plot the maximal errors over different regions D.Figure (a) plots the maximal errors along the line considered in figure 7. The trapezoid errors, in black,are large and do not decay as n increases. They are irregular since the relative distance ah to thenearest point vortex changes irregularly as n increases. The correction presented in section 2, labeled“Correction 1” and shown in blue, clearly decays as O(h2). The third order correction presented laterin section 4, labeled “Correction 2” and shown in red, clearly decays as O(h3). Figure (b) plots themaximal errors away from the plate edges, for −0.8 < x < 0.8. The blue and red data clearly decay asO(h2) and O(h3), respectively. Figure (c) plots the maximal errors everywhere and captures the large


10-3

10-2

1/n

10-10

10-7

10-4

10-1

102

Maxim

um

err

or

(a)

10-3

10-2

1/n

10-10

10-7

10-4

10-1

102

Maxim

um

err

or

(b)

10-3 10-2

1/n

10-10

10-7

10-4

10-1

102

Maxim

um

err

or

(c)

Trapezoid

Correction 1

slope 2

Correction 2

slope 3

Figure 12. Maximal errors over all points on a grid over (a) the line x = −0.753, (b) regionaway from tip, |x| ≤ 0.8, (c) the whole domain shown. Errors for n = 50, 100, 200, 400, 800meshpoints are shown. The error that is maximized is the minimum of the relative error and theabsolute error.

relative errors near the plate edges. They also decay as O(h2) and O(h3) but with larger constant ofproportionality.

3.1.3. Effects of finite precision as d → 0. The above results show that in this sample application theasserted convergence rates hold, and in theory, arbitrarily high accuracy can be obtained. However, theeffects of finite machine precision are larger than one may expect. The method consists of adding acorrection to a bad result. In case of the point vortex approximation, the bad error can be arbitrarilylarge, depending on the closeness of the target point to the point vortices. In that case a correction thatis accurate to 14 digits can only gain 14 digits of accuracy, which may not be enough to obtain smallerrors. For this reason, the data point in figure 12(b) for n = 800 for the O(h3) method was computedwith quadruple precision, even though the actual final maximal error is only 10−9. While the numericalimplementation of the method can be improved to reduce the accumulation of roundoff error, one needsto be aware of this issue.

A simple solution is to note that the erroneous velocities induced by the point vortex flow at distanced have magnitude uv ≈ ∆Γ/(2πd) where ∆Γ = hΓ′(k0) is the circulation of the nearest vortex. WhenΓ′ = O(1), uv ≈ h/(2πd), and the small distances at which unacceptably large errors occur can beestimated. For example, figure 11 was obtained with d/h as small as 5 · 10−4. Thus the predictedmaximal trapezoid errors on the grid are approximately constant for all n, of order 1/(2π ·0.0005) ≈ 300.This is consistent with the results shown in figure 12. For such small distances interpolation could beused to yield good accuracy and avoid the effects of finite machine precision.

3.2. Potential flow past two cylinders. Our second example consists of potential flow past twocylinders. The cylinders have radius 1, are separated by a distance 2A, centered at (0, 1 ± A), in afar field background flow U∞ = 〈1, 0〉. The two bounding vortex sheets are parametrized by x±(α) =〈cosα, sinα± (1 +A)〉, α ∈ [0, 2π] and discretized by equally spaced points

(18) x+(αk) , x−(αk + ∆α) , αk = kh , h = 2π/n , k = 0, . . . , n

Here, ∆α 6= 0 adds a small rotation to the points on the bottom cylinders. Generically, the discretizationof nearby objects is not necessarily symmetric. In the present example, perfect symmetry ∆α = 0 wasfound to improve the results significantly. This was was also observed by Ellington [?], who, for thisreason, uses symmetric points to perform symmetric vortex sheet calculations. Here we consider genericnonsymmetric distributions and use a representative value ∆α = h/4. Figure 13(a) shows the resultingnonsymmetric point vortex positions for n = 50.

Figure 13(b,c) shows the computed sheet strength along the top cylinder for the case of two cylindersseparated by half-distance A = 0.001. It shows a closeup of Γ′(αk) near α = 1.5π, which is thepoint on the top cylinder closest to the bottom cylinder. The sheet strength at a point x0 on the topcylinder depends on the velocity induced by the bottom cylinder, as well as the background flow. Infigure (b) the induced velocity is computed by the trapezoid rule. For x0 near the bottom cylinder,


x0

(a)

x

y

1.4 1.45 1.5 1.55 1.6

/

-10

0

10

20

30

40

50

'()

(b)n=50

100

200

400

800

1600

1.4 1.45 1.5 1.55 1.6

/

-10

0

10

20

30

40

50

'()

(c)n=50

100

200

400

800

1600

Figure 13. Flow past two cylinders of radius R = 1 separated by distance 2A. (a)Sketch showing discretization of boundary, nonsymmetric with αoff = h/4. (b,c) Com-puted circulation distribution Γ′(α) along the top cylinder, for A = 0.001 and αoff = h/4,using the indicated values of n, with (b) trapezoid rule, (c) corrected quadrature.

the induced velocity is computed inaccurately, leading to large errors in the resulting sheet strength.Figure (b) shows no apparent convergence in the sheet strength for n ≤ 1600. Figure (c) shows theresults obtained when the induced velocity is instead computed using the corrected quadrature. Clearconvergence is observed. The values with n = 100 are already about as good as n = 1600 in figure (b).These results are computed with the O(h3) method, although they are, to the eye, indistinguishablefrom the O(h2) results (not shown).

The sheet strength shown in (c) converges fast. For example, the error in the maximum value atα = 1.5π, starting with n = 100, decreases by factors 3.8, 10.3, 32.0 every time n is doubled. The valueat α = 1.52π starting with n = 50, decreases by factors 1.5, 8.0, 64.3, 111.3, every time n is doubled.

Figure 14 plots streamlines for the flow past the two cylinders for half-distance A = 0.01, where bothsheet strength and streamlines are computed with n = 50 and (a) the trapezoid rule, (b) the correctedO(h3) quadrature. The result in (a) shows lack of symmetry about both x = 0 and y = 0, and manystreamlines traversing the cylinders. The result in (b) recovers all expected symmetries and has nostreamlines crossing the boundary. The only irregularity is a small blib near the origin. This blib isslightly more noticeable with the O(h2) method (not shown). Figure 15 (a,b) shows a closeup of thestreamlines shown in figure 14 near the origin, and more clearly shows the shape of the blib in (b) inthe streamlines closest to the boundary. As reference, the n = 50 gridpoints are indicated. The size ofthe gap between the cylinders is about 1/6 the distance between gridpoints. Figure 15 (c,d) shows thestreamlines computed with n = 100, showing that the blib in figure (b) is resolved.

These results show that the corrected quadrature, by accurately recovering the velocity induced bythe nearby cylinder, yields accurate sheet strength and streamlines.

4. Convergence as d→ 0

Here we show that the method presented in this paper to compute near-singular integrals of the form(1) converges as d→ 0. In particular, we show that for points near the boundary, the integration errorusing approximation (10-12) is O(hd) + O(h4), where d is the distance from the target point to theboundary and h is the meshsize used for the trapezoid rule. It therefore follows that for d ≤ h themaximal errors near the boundary are of order O(h2), and that the error vanishes uniformly as d→ 0,up to a small O(h4) term independent of d. These results are consistent with the sample numericalresults presented in section 3.

The accuracy near the boundary can be improved as desired using higher order approximations. Toillustrate, the next section also presents a method for G1 of order O(hd2, h4), used in some of the resultsshown earlier. In this case, the maximum errors are O(h3). The O(h4) term can be reduced using higherorder generalizations of the trapezoid rule, which would be specially simple in periodic problems.


Figure 14. Potential flow past two cylinders of radius 1, separated by half-distanceA = 0.01, computed with n = 50 points on each cylinder, using (a) the trapezoid ruleand (b) the corrected quadrature.

(a)

-0.2

0

0.2

y

(b)

-0.2

-0.1

0

0.1

0.2

(c)

-0.5 0 0.5

x

-0.2

0

0.2

y

(d)

-0.5 0 0.5

x

-0.2

-0.1

0

0.1

0.2

Figure 15. (a,b) Closeup of figure 14, using n = 50. (c,d) Same as (a,b), but using n = 100.

4.1. Summary of approach and main results. The method consists of approximating∫

(G−H) byT4[G−H], where H(α) is an approximation of G(α). To derive the order of convergence we thus needto estimate E4[G−H], for each G = G1, G2 or G0. The approximations H are obtained by truncatinga Taylor series expansion of G about α = αp. All the terms in these Taylor expansions are of one of theforms

Hjk =αk

(d2 + c2α2)j, j ≥ 1, k ≥ 0 ,(19a)

H0k = αk log(d2 + c2α2), k ≥ 0 ,(19b)

where α = α − αp. Below we derive upper bounds for the error in integrating each of these basisfunctions. That is the main component of the argument. The order of the approximation E4[G−H], as


well as higher order approximations, directly follow from these bounds. Here, we summarize the mainresults. The derivation of the bounds is given in the remaining subsections.

We note that c2 is always bounded away from 0 when d is sufficiently small, as argued earlier (see alsoAppendix A), and thus it is sufficient to consider c2 = 1 for the arguments below. As shown in sections4.2-4.4, the errors E4[Hjk] =

∫I Hjk−T [Hjk] over an interval containing the gridpoint αk0 closest to αp

satisfy the following bounds as d→ 0: for all j ≥ 1,

(20a) E4[Hjk] = E4

[αk

(d2 + α2)j

]= O(hdk−2j) ,

if k ≤ 2j + 3. For larger k ≥ 2j + 4, an O(h4) needs to be added to the right hand side. For j = 0,

(20b) E4[H0k] = E4[ αk log(d2 + α2) ] =

O(h log d) , if k = 0,

O(hk+1) , if 0 < k ≤ 3

O(h4 log h) , if k ≥ 4.

As a result we obtain the following ordering of the terms in the Taylor series expansion of G1:

G1(α) =F1(x(α),x0)

|x(α)− x0|2ω(α) =

c0d+ c1α+ c2α2 + c3α

3 +O(α4)

d2 + c2α2 + eα3 + e4α4 +O(α5)

=c0d+ c1α+ c2α

2 + c3α3 +O(α4)

d2 + c2α2

[1− eα3 + e4α

4 +O(α5)

d2 + c2α2+e2α6 +O(α7)

(d2 + c2α2)2+

O(α9)

(d2 + c2α2)3

]

=

H1(α)︷︸︸︷c0d+ c1α

d2 + c2α2︸︷︷︸E4[·]=O(h/d)

+c2α

2

d2 + c2α2− e c0dα

3 + c1α4

(d2 + c2α2)2︸︷︷︸E4[·]=O(h)

+

H11︷︸︸︷c3α

3

d2 + c2α2− e c2α

5

(d2 + c2α2)2− e4

c0dα4 + c1α

5

(d2 + c2α2)2+ e2

c0dα6 + c1α

7

(d2 + c2α2)3︸︷︷︸E4[·]=O(hd)

+O(α4)

d2 + c2α2+

O(dα5, α6)

(d2 + c2α2)2+

O(dα7, α8)

(d2 + c2α2)3+O(dα9, α10)

(d2 + c2α2)4︸︷︷︸E4[·]=O(hd2,h4)

(21)

The order of the error in integrating each of the terms by T4 is indicated below the underbraces.The first overbrace indicates the function H1 whose integral is evaluated exactly. We conclude thatthe error E[G1 − H1] = O(hd, h4). The second overbrace indicates the function H11 containing thenext order terms. It follows that the error E[G1 − (H1 +H11)] = O(hd2, h4). This describes the thirdorder method used in some of the results in section 3. The third order method thus requires adding5 additional corrections c′jkE[Hjk] stemming from H11 to the 5 corrections stemming from H1, in step

2(e) of the method (see section 2.2). The third order method does require the fourth derivative at αpon the boundary, since e4 = (xp − x0) ·

....x p/12 + xp ·

...xp/3 + xp · xp/4. If the boundary is given by

a fixed object such as in the examples presented in section 2, that is not a problem. The third ordermethod also requires the third derivative of the density ω(α), via c3, which may be less desirable whenthe density has large variations.

For the results given in equation (21) it was important to remember that the constant term in thenumerator is proportional to d, c0 = c0d, since the numerator F1 has a simple root at x0 = xp, asargued earlier in section 2.2. The results for G2, presented next, require the scaling of the first three


coefficients in the numerator, c0 = c0d3, c1 = c1d

2, c2 = c2d. Throughout, the implication is that allconstants denoted by a tilde are O(1). The scaling holds since the numerator F2 must have a triple rootat x0 = xp. We thus obtain the following ordering of the terms in G2:

G2(α) =F2(x(α),x0)

|x(α)− x0|4ω(α) =

c0d3 + c1d

2α+ c2dα2 + c3α

3 + c4α4 +O(α5)

(d2 + c2α2 + eα3 +O(α4))2

=c0d

3 + c1d2α+ c2dα

2 + c3α3 + c4α

4 +O(α5)

(d2 + c2α2)2

[1− 2

eα3 +O(α4)

d2 + c2α2+

O(α6)

(d2 + c2α2)2

]

=

H2(α)︷︸︸︷c0d

3 + c1d2α+ c2dα

2 + c3α3

(d2 + c2α2)2︸︷︷︸E[·]=O(h/d)

+c4α

4

(d2 + c2α2)2− 2e

c0d3α3 + c1d

2α4 + c2dα5 + c3α

6

(d2 + c2α2)3︸︷︷︸E[·]=O(h)

+O(α5)

(d2 + c2α2)2+O(d3α4, d2α5, dα6, α7)

(d2 + c2α2)3+O(d3α6, d2α7, dα8, α9)

(d2 + c2α2)4︸︷︷︸E[·]=O(hd,h4)

(22)

Thus, also E[G2 −H2] = O(hd, h4).For G0 we obtain:

G0(α) =ω(α) log |x− x0|2 =(c0 + c1α+ c2α

2 +O(α3))

log(d2 + c2α2 + eα3 +O(α4)

)=[c0 + c1α+ c2α

2 +O(α3)][

log(d2 + c2α2) +eα3 +O(α4)

d2 + c2α2+

O(α6)

(d2 + c2α2)2]

=

H0(α)︷︸︸︷c0 log(d2+ c2α2)︸︷︷︸E4[·]=O(h log d)

+ c1α log(d2+ c2α2)︸︷︷︸E4[·]=O(h2)

+ c2α2 log(d2+ c2α2)︸︷︷︸E4[·]=O(h3)

+

H01(α)︷︸︸︷e

c0α3

d2 + c2α2︸︷︷︸E4[·]=O(hd)

+O(α4)

d2 + c2α2+

O(α6)

(d2 + c2α2)2︸︷︷︸E4[·]=O(hd2)

+O(α3) log(d2+ c2α2)︸︷︷︸E4[·]=O(h4 log h)

.

(23)

Thus we conclude E4[G0−H0] = O(hd, h4). Note also that E4[G0−(H0+H01)] = O(hd2, h4). In view ofthe simple form of H01, the resulting third order method is easy to implement. Furthermore, since e = 0for circular boundaries as in the two cylinder example in Section 2, the second order approximation ofthe streamfunction for a given ω(α) is automatically of third order.

This concludes the presentation of the main results regarding the convergence of the method intro-duced in this paper. The remaining sections are devoted to demonstrating equation (20) on which theseresults are based. We first address the proper fractions Hjk with j ≥ 1, k < 2j, then generalize theresult to improper fractions k ≥ 2j, and lastly consider the logarithmic functions H0k.

4.2. Proper fractions Hjk, j ≥ 1, k < 2j. Our first goal is to bound E4[H] =∫I H − T4[H], where

H = Hjk is a proper fraction, k < 2j. To begin, note that it is sufficient to consider the standard secondorder trapezoid error over a small interval I = [αk0−1, αk0+1] with only three gridpoints, containing thepoint αk0 closest to αp. For small d/h, this error captures the leading order contribution to E4[H] overany larger interval, since the near-singular behaviour of H occurs in an interval of width O(d) centered


0 0.5 1 1.5 2 2.5 3

ah/d

-E[H

10]

d2/h

-0.2

1.2(d) h/d=10

h/d=100

h/d=1000

h/d=10000

(0,1)

0 0.5 1 1.5 2 2.5 3

ah/d

E[H

11]

d/h

0

0.5 (e)

h/d=10

h/d=20

h/d=40

h/d=80

(1,1/2)

0 0.5 1 1.5 2 2.5 3

ah/d

-E[H

22]

d2/h

-0.2

0.3(f)

h/d=10

h/d=100

h/d=1000

h/d=10000

(1,1/4)

Figure 16. Error E[f ] over I = [(k0 − 1)h, (k0 + 1)h], for (a,b) f1, (c,d) f2, (e,f) f3.The left column shows the error, represented by the shaded area, for sample values ofa, d, h. The right column plots scaled values of E[f ] for a range of a, d, h.

at αp. We denote the standard trapezoid error over I by E[H] =∫H − T , where

(24a) T = T [H] =h

2[H(αk0−1) + 2H(αk0) +H(αk0+1] .

and

(24b)

∫H =

∫ αk0+1

αk0−1

H(α) dα .

Furthermore, as argued above, it suffices to consider c2 = 1.The approach to bound E[H] becomes clear by considering some examples. The three sample func-

tions H10, H11, H22 are plotted in figure 16(a,b,c) for generic values of d. Their maximum values andcharacteristic lengthscales are indicated. Like all functions Hjk, they have a self-similar shape

(25) Hjk =αk

(d2 + α2)j=

1

d2j−kuk

(1 + u2)j=

1

d2j−kgjk(u)

where u = α/d. For k < 2j, gjk(u) is a bounded function that vanishes as u → ±∞. As a result, themaximum value of |H| satisfies

(26) |Hjk|max =|gjk|maxd2j−k

= O(dk−2j) ,

Also, characteristic widths of the near-singular behaviour are O(d). For example in figure (a), 4d is thewidth of the function at the level H10 = |H10|max/5. In figure (c), 3d is the width of the function atthe level H22 = 4|H22|max/13. In figure (b), d is the half-distance between the extrema of H11.

Figures 16(a,b,c) also show the three gridpoints αk0−1, αk0 , αk0+1, equally spaced by distance h. Theshaded area represents the trapezoid error E[H] =

∫H − T over the chosen interval I = [αk0−1, αk0+1].

As we saw earlier, the error depends not only on the meshsize h and the distance d, but on the distanceah of αp to the closest gridpoint αk0 (see figure 16b). Here a ∈ [−1/2, 1/2), where positive a correspondsto αp > k0h, and negative a corresponds to αp < k0h. We need to bound E[H] over all values of ah.By symmetry of all basis functions Hjk it is enough to consider a > 0.


In all three figures 16(a,b,c) the value of ah ≥ 0 is chosen so that the maximum of |H| is taken on atthe middle gridpoint αk0 . This choice of ah is crucial for our argument. The key point is to recognizethat if d� h, then:(1) The trapezoid approximation T is largest in magnitude when ah is such that the maximum of|H| is taken on at the middle gridpoint, αk0 . In that case, the value of T , given in (24a), isdominated by the contribution of the middle gridpoint, h|H(αk0)| = h|H|max. That is,

(27a) |T |max ≈ h|H|max ,where |T |max = max

a∈[−1/2,1/2)|T |.

(2) The maximal trapezoid approximation is much larger in magnitude than the integral of H

(27b) |T |max �∣∣∣∣∫ H

∣∣∣∣ ,where

∫H is given in (24b).

Since |E[H]| = |∫H − T | ≤ |

∫H|+ |T | ≤

∫|H|+ |T |max, it follows from (27b) that |T |max dominates

the error E[H]. From (27a) it then follows that

(28) |E[H]|max ≈ |T |max ≈ h|H|max ,

and that this maximum is taken on when ah is such that |H| is maximal at αk0 . Finally, using thescaling (26) for Hmax we obtain the desired bound:

(29) E[Hjk] = O(hdk−2j) .

Item (1) follows from equation (24a) and the self-similar nature of H: the middle term in (24a) islargest when H is maximized at αk0 , and the relative values of H/Hmax at any other grid point αkvanish as d → 0. Item (2) can be deduced from the figures: for example, if H is even, as in figures16(a,c), |T |max approximately equals the area of the triangular region under the two slanted lines, whichis clearly bigger than the area of the white region under the curve y = H(α) for small d/h. If H is odd,as in figure 16(b), then

∫H ≈ 0 by symmetry, and |T | ≈ h|H|max � 0.

To illustrate these results, figures 16(d,e,f) plot the actual errors E[H] for the three sample functionsH10, H11, H22, scaled by the predicted behaviour (29), as functions of ah/d, for a range of values ofa, d, h. The red dots represent the predicted approximate maximum values of E[H] and the value ofah/d at which it ocurrs. For example, for H10(α), shown in figure 16(a), the maximum is 1/d2 and takenon at α = 0. Thus |E[H10d

2/h|max ≈ 1, taken on when ah/d = 0. For H11(α), shown in figure 16(b), themaximum is 1/2d, taken on at α = d. Thus |E[H11d/h|max ≈ 1/2, taken on when ah/d = 1. The figuresshow that as h/d → ∞, the actual error, scaled appropriately, quickly converges to a limiting curvewith magnitude closely bounded by the ordinate of the red dot. These results illustrate the accuracy ofthe approximation (28) and are consistent with the main conclusion (29). We remark that the curvesshown in figures (b,d,f) capture the scaling and the dependence of the error on all three parameters h, dand a.

These results remain unchanged if T is replaced by T4, since the contributions of the endpointderivatives to T4 are insignificant for small d/h.

4.3. Nonproper fractions Hjk, j ≥ 1, k ≥ 2j. The result for non-proper fractions is obtained fromthe above result (29), now switching from T to T4, using long division and the fact that polynomials of


0 0.5 1 1.5 2

ah/d

E[H

00]/

A0

0.96

1.01(d)

h/d=1010

h/d=1020

h/d=1040

h/d=1080

(0,1)

0 0.1 0.2 0.3 0.4 0.5

a

E[H

01]/

h2

-0.2

0.3(e)

h/d=10

h/d=20

h/d=40

h/d=80

0 0.1 0.2 0.3 0.4 0.5

a

E[H

02]/

h3

0.15

-0.15

(f)

h/d=10

h/d=20

h/d=40

h/d=80

Figure 17. (a,b,c) Graphs of the bounded components Hb0k for k = 0, 1, 2, for generic values

of h, d. The value of a is chosen to maximize the trapezoid error, which is represented by theshaded area. Characteristic length scales are indicated. (d,e,f) Scaled errors E[H0k], k = 0, 1, 2,for a range of values of a, h, d. In (d), A0 = 2h(log(d/h)+2)+2dπ approximates the shaded area

in (a).

degree p ≤ 3 are integrated exactly by the trapezoid rule T4. For example,

H12(α) =α2

d2 + α2= 1− d2 1

d2 + α2, ⇒ E[H12] = O(h) ,(30a)

H13(α) =α3

d2 + α2= α− d2 α

d2 + α2, ⇒ E[H13] = O(hd) ,(30b)

H24(α) =α4

(d2 + α2)2=

α2

d2 + α2− d2 α2

(d2 + α2)2, ⇒ E[H24] = O(h) ,(30c)

H25(α) =α5

(d2 + α2)2=

α3

d2 + α2− d2 α3

(d2 + α2)2, ⇒ E[H25] = O(hd) .(30d)

These results can be generalized using induction to obtain that for all k ≥ 0,

(31) E4

[αk

(d2 + α2)j

]= O(hdk−2j) ,

if k ≤ 2j + 3. For larger values of k, a generic error term O(h4) needs to be added to the right handside.

4.4. Logarithmic terms. The logarithmic terms H0k are treated similarly, but a few further detailsare needed. For k = 0 we rewrite

(32) H00(α) = log(d2 + α2) = log(1 + α2)︸︷︷︸Hu

00

+ log

(d2 + α2

1 + α2

)︸︷︷︸

Hb00

,

where the first term, Hu00, is independent of d and captures the unbounded behaviour of H, while the

second term, Hb00, is bounded, with maximum value |Hb

00|max = 2 log d, and approaches 0 as α→ ±∞.Although Hb = Hb

00 is not self-similar, it is approximately self-similar. For example Hb = Hbmax/4 when

α ≈ ±d1/4, as indicated in figure 17(a). The same arguments as before therefore apply and result in


a bound for the second term in (32), E[Hb00] ≈ 2h log d = O(h log d). The first term is regular and is

integrated to O(h4) by T4. The logarithmic term dominates the total error, so

(33) E4[ log(d2 + α2) ] = O(h log d) .

The bounded term Hb00, and the total scaled error E[H00] for a range of values a, d, h are plotted in

figures 17(a,d), showing results consistent with the arguments just made.For k ≥ 1, we rewrite H0k as in the following examples:

H01(α) = α log(d2 + α2) = α log(α2) +d [ u log(1 + 1/u2) ] ,(34a)

H02(α) = α2 log(d2 + α2) = α2 log(α2) + d2 +d2[u2 log(1 + 1/u2)− 1] ,(34b)

H03(α) = α3 log(d2 + α2) = α3 log(α2) + d2α +d3[u3 log(1 + 1/u2)− u] ,(34c)

H04(α) = α4 log(d2 + α2) = α4 log(α2)︸︷︷︸Hu

0k

+ [d2α2 − d4/2]︸︷︷︸Hp

0k

+ d4[u4 log(1 + 1/u2)− u2 + 1/2)]︸︷︷︸Hb

0k

,(34d)

where u = α/d. Here, the last term, the bounded term Hb0k, is self-similar, and approaches 0 as

α→ ±∞. It is obtained by subtracting a polynomial Hp0k, given by the middle term, that describes the

asymptotic behavior of the logarithmic term. The asymptotic polynomial can be found for any k usingTaylor expansions of log(1 + 1/u2) as u→ ±∞. The first term is unbounded and independent of d.

The bounded self-similar term has maximum |Hb0k|max = O(dk) and thus, by the same arguments as

above,

(35) E[Hb0k] = O(hdk) .

The middle term is a polynomial integrated exactly by T4 for all degrees ≤ 3. The first term howeverdominates the integration error. First, we note that the results of Sidi and Israeli [15] for functions ofthe form αk log α2 do not apply here, since their results assume that the logarithmic singularity at αpfalls on a gridpoint, which is not the case here. Instead we treat the first term as follows:

(36) E4[αk log(α2)] = hkE4,I [(α/h)k log(α/h)2] + E4[α

k log h2] = hk+1E4,I/h[uk log u2] = O(hk+1)

for k ≤ 3. The result (36) follows by a change of variables u = α/h since the interval I is proportionalto h. We also used the fact that the polynomial αk log h2 is integrated exactly for k ≤ 3. For k > 3,this term leads to an additional O(h4 log h) term that needs to be added to the total error.

The arguments above show that for k > 0

(37) E4[ αk log(d2 + α2) ] = O(hk+1, h4 log h) .

The scaled bounded components Hb0k, k = 1, 2, and the total scaled error E4[H0k] for a range of values

a, d, h are plotted in figures 17(b-c,e-f), showing consistency with the arguments just made.This completes the derivation of equation (20) and of the convergence of the method.

5. Summary

This paper presents a method to evaluate boundary integrals of the types given in (1), in the casewhen the target point x0 is not on the boundary but at a small distance d from it. These are the typesof integrals that describe the solution of elliptic boundary value problems. In particular, they arise invortex sheet flow and in Stokes flow, in either planar or axisymmetric geometries. The integrals arenear-singular when d is small and are difficult to compute accurately with standard quadrature rules.

The proposed method consists of approximating the integrands G by a function H that captures thenear-singular behaviour and can be integrated exactly. The approximation is obtained by truncating aTaylor series expansions of the integrand about a basepoint on the domain boundary. The truncationconsists of a sum of basis functions Hjk. The numerical method amounts to adding a correction E[H]for each of the basis functions to the trapezoid approximation T4[G] of the original integral. Analyticalbounds are obtained for the integration error of each basis function. These bounds determine thenumber of corrections required for the desired order of accuracy.

Two versions of the method, of order O(hd, h4) and O(hd2, h4), respectively, are listed explicitely,where d is the distance of the target point to the boundary and h is the meshsize used for the underlying


trapezoid rule. The resulting integration error in each case vanishes uniformly as d → 0 and hasmaximum value at a distance d ∼ h from the boundary that is of order O(h2) and O(h3) respectively.The methods were applied to two examples of planar vortex sheet flow consisting of potential flow past aplate and past two cylinders. Convergence of the numerical results at the analytically predicted rates isconfirmed. The method is conceptually simple and simple to implement, with a cost of O(1) operationsper target point, and is presented here as a valuable alternative to resolve near-singular integrals.

Appendix A. Positivity of c2

This section shows that the quantity

(38) c2 = |xp|2 + xp · (xp − x0)

is positive in the cases considered in this paper, namely a flat boundary with nonuniform point distri-bution and curved boundaries with equally spaced points. For curved boundaries with target points onthe “inside”, defined below, d must be sufficiently small.

In the flat case, see figure 18(a), xp is tangent to the plate while xp − x0 is normal to it, thusxp · (xp − x0) = 0 and

(39) c2 = |xp|2 > 0

for all d, and xp in the plate interior.In the equally spaced case, |xp| = sα = constant, where s is arclength and sα = ds/dα. We consider

two possibilities: either x0 lies on the “outside” of the curve, as in figure 18(b), or x0 lies on the “inside”of the curve, as in figure 18(c). Here, the “outside” is defined as the side opposite to that pointed intoby the normal vector N at xp, where

(40) N =1

κ

dT

ds, T =

dx

ds,

while the “inside” is the side containing the osculating circle of the curve at xp. Here, κ = |dT/ds| isthe curvature at xp. Note that in the equally spaced case,

(41) T =xpsα

, N =dT

ds/

∣∣∣∣dTds∣∣∣∣ =

1

κ

xps2α

.

Thus if x0 is on the outside, xp − x0 = dN and xp · (xp − x0) = dκs2α > 0 so

(42) c2 = s2α(1 + dκ) > s2α = |xp|2 > 0 ,

for all d. If x0 is on the inside, x− x0 = −dN and xp · (xp − x0) = −dκs2α. Therefore

(43) c2 = s2α(1− dκ) >1

2s2α =

1

2|xp|2 > 0 ,

Figure 18. Sketch showing different scenarious: (a) flat curve, and (b,c) uniform ar-clength parametrization, sα =constant, with x0 outside of the curve in (b) and inside ofthe curve in (c).


provided d < 1/(2κ) = Rosc/2, where Rosc = 1/κ is the radius of the osculating circle.

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