## 1 Introduction

The method of fundamental solutions, or the charge simulation method, [6, 15] is a fast numerical solver for potential problems

(1) |

where is a domain in the -dimensional Euclidean space ,
and is a function given on .
In two dimensional problems , equalizing the Euclid plane
with the complex plane ,
the method of fundamental solutions gives an approximate solution of
(1) of the form
^{2}^{2}2The scheme shown here is the invariant scheme [16, 17] proposed by Murota.

(2) |

where are unknown real coefficients such that

(3) |

and are points given in . We call the “charges” and the “charge points”. We remark that the approximate solution exactly satisfies the Laplace equation in . Regarding the boundary condition, we pose the collocation condition on , namely, we assume that satisfies the equations

(4) |

with given on the boundary , which are called the “collocation points”. The equations (4) are rewritten as

(5) |

which, together with (3), form a system of linear equations with respect to . We determine the unknowns by solving the linear system (3) and (5) and obtain the approximate solution . The method of fundamental solutions has the advantages that it is easy to program, its computational cost is low, and it shows fast convergence such as exponential convergence [11] under some condition. It was first used for studies of electric field problems [25, 26], and now it is widely used in science and engineering.

The method of fundamental solutions is also used for the approximation of complex analytic functions. The idea of the method of fundamental solutions is applied to various problems other than potential problems, for example, problem of scattering of earthquake waves [24]. Let be an analytic function in a domain . The real part of , which is a harmonic function in , can be approximated using the form (2), and the imaginary part of , which is the conjugate harmonic function of , is approximated using

Then, the analytic function is approximated using a linear combination of the complex logarithmic functions

(6) |

From this point of view, Amano applied the method of fundamental solution to numerical conformal mappings [1, 2].

In this paper, we examine the problem of two-dimensional potential flow past an infinite doubly-periodic array of obstacles as shown in Figure 1. A two-dimensional potential flow is characterized by a complex velocity potential [14], which is an analytic function in the flow domain such that it gives the velocity field by , and its imaginary part satisfies the boundary condition

(7) |

Physically, the boundary condition (7) means that the fluid flows along the boundary since the contour lines of are the streamlines. Therefore, in order to obtain the potential flow of our problem, we have to find an analytic function in satisfying the boundary condition (7).

We have, however, one problem in applying the method of fundamental solutions to our problem. The velocity field is obviously a doubly-periodic function due to the the double periodicity of the flow domain . Then, the complex velocity potential involves a doubly-periodic function, and it is difficult to approximate it using the form (6) of the conventional method of fundamental solutions. To overcome this challenge, we propose to solve our problem by a method of fundamental solutions using a doubly-periodic fundamental solution, that is, a logarithmic potential with a doubly-periodic array of sources. We construct a doubly-periodic fundamental solution using the theta functions and approximate the complex velocity potential by a linear combination of the doubly-periodic fundamental solutions.

The previous works related to this paper are as follows. As studies of problems of periodic flow, Zick and Homsy [27] proposed an integral equation method for three-dimensional Stokes flow problems with a three-dimensional periodic array of spheres, where the solution is given by an integral including the periodic fundamental solution. Greengard and Kropinski [7] proposed an integral equation method for two-dimensional Stokes flow problems in doubly-periodic domains, where an approximate solution is given as a complex variable formulation and the fast multipole method is used. Liron [13] studied Stokes flow due to infinite array of Stokeslets and applied it to the problems of fluid transport by cilia. As studies on methods of fundamental solutions applied to problems with periodicity, Ogata et al. proposed a method of numerical conformal mappings of complex domains with a single periodicity [22], where the mapping function, an analytic function involving a singly-periodic function, is approximated by the method of fundamental solution using the singly-periodic fundamental solutions. They proposed a method of fundamental solutions also to the problems of two-dimensional periodic Stokes flows [18, 19], where the solutions are approximated by the periodic fundamental solutions of the Stokes equation, that is, the Stokes flows induced by a periodic array of point forces. The author proposed a method of fundamental solutions for the two-dimensional elasticity problem with one-dimensional periodicity [23], where the solution is approximated using the periodic fundamental solutions of the elastostatic equation, that is, the displacements induced by concentrated forces in a periodic array.

The remainder of this paper is structured as follows. Section 2 proposes a method of fundamental solutions for our problem of potential flow with double periodicity. Section 3 shows some numerical examples which show the effectiveness of our method. In Section 4, we conclude this paper and present problems related to future studies.

## 2 Method of fundamental solutions

We consider a potential flow past the doubly-periodic array of obstacle. The flow domain is mathematically given by

where is one of the obstacles, a simply-connected domain and

with complex numbers and giving the periods of the obstacle array such that . We assume that the spatial average of the velocity field is a uniform flow, in whose direction we take the real axis, when the flow is observed on the direction of . Mathematically, this assumption is formulated as

(8) |

where is the magnitude of the velocity of the unit flow and is the line segment connecting two points and

is the unit normal vector on

.As mentioned in the previous section, a two-dimensional potential flow is characterized by a complex velocity potential, , which is an analytic function in the flow domain such that it gives the velocity field by

(9) |

and its imaginary part is constant on the boundary of the domain . The condition (8) that the average of the flow is uniform is summarized as

(10) |

Therefore, our potential flow problem is mathematically reduced to the problem to find an analytic function in satisfying

(11) |

and (10).

In solving our problem by the method of fundamental solution method, however, we have one problem. The complex velocity potential involves a periodic function since the velocity field given by (9) is obviously a periodic function, and it is impossible to approximate by the conventional method of fundamental solutions using the form (2). We propose to approximate the complex velocity potential by

(12) |

where is the theta function [3]
^{3}^{3}3
In [3], the theta function is defined by

with

are points given in and are unknown real coefficients such that

(13) |

We call the “charge” and the “charge points”. The theta function is an entire function which has simple zeros at , and satisfies the pseudo-periodicity

(14) |

Therefore, the functions
appearing on the right-hand side of (12) can be regarded
as the complex logarithmic potential with sources at the points
and it is a periodic fundamental solution of the two-dimensional Poisson equation
^{4}^{4}4
Hasimoto [9] presented a periodic fundamental solution of the two-dimensional
Poisson equation in terms of the Weierstrass elliptic functions
.

The approximate velocity potential satisfies the condition (10). In fact,

where we used the first formula of (14) on the third equality and (13) on the fourth equality.

The approximate velocity potential satisfies the pseudo-periodicity

(15) |

In fact, the first equation is proved above, and the second equation is proved similarly to the first equation using the second formula of (14). Therefore, the approximate velocity given by is a doubly-periodic function, namely,

It means that the approximate complex velocity is an elliptic function of periods .

The approximate velocity potential in (12) is an analytic function in such that it satisfies the condition (10) and gives velocity field which is a doubly-periodic function of periods . Regarding the boundary condition (11), we pose a collocation condition on , namely, we assume the equation

(16) |

where are given points on called the “collocation points” and is an unknown real constant. The equations (16) are rewritten as

(17) |

The equations (13) and (17) form a system of linear equations with respect to the unknowns and . We determine the unknown charges by solving this linear system and obtain the approximate velocity potential . Owing to the pseudo-periodicity (15), the approximate potential approximately satisfies the boundary condition on the boundary of other obstacle , , that is,

## 3 Numerical examples

In this section, we show some numerical examples which show the effectiveness of our method. All the computations were performed using programs coded in C++ with double precision.

x y | x y |

x y | x y |

Figure 2 shows the examples of flows past a doubly-periodic array of cylinders, that is, flows in the domain

where | |||

with a positive constant . The charge points and the collocation points are respectively taken as

(18) |

where is a constant such that and was taken as in the examples shown in Figure 2

. To estimate the accuracy of our method, we evaluated the value

where is the constant obtained in solving the system of linear equations (13) and (16). The value shows how accurately the approximate potential satisfies the boundary condition (7). Figure 3 shows evaluated for the above numerical example with and for , which appears in giving the charge points by (18). The figure shows that the approximation of our method converges exponentially as the number of unknowns increases.

## 4 Concluding Remarks

In this paper, we proposed a method of fundamental solutions for the problems of two-dimensional potential flow past a doubly-periodic array of obstacles. In terms of mathematics, our problem is to find the complex velocity potential, an analytic function in the flow domain with double periodicity. The solution obviously involves a doubly-periodic function, and it is difficult to approximate it by the conventional method of fundamental solutions. Then, we proposed a new method of fundamental solution for this problem using the periodic fundamental solutions, which is the logarithmic potential with a doubly-periodic array of sources and contains the theta functions. The numerical examples showed the effectiveness of our method.

We have two issues regarding this paper for future study. The first problem is a theoretical error estimate of our method. Theoretical studies on the accuracy of the method of fundamental solution have been presented for special cases such as two-dimensional Laplace equation and Helmholtz equation in a disk [11, 4, 20] and two-dimensional Laplace equation in a domain with an analytic boundary [12, 21]. However, theoretical error estimate still remains unknown for many types of problems and methods including our method. The author believes it one of the most important works on the method of fundamental solutions to give a theoretical error estimate.

The second problem is to extend our method to other problems with periodicity than two-dimensional potential problems. The author has already given methods of fundamental solutions for Stokes flow problems [19, 18] and elasticity problems [23] with periodicity. In the work on periodic Stokes flow [18], the author et al. proposed to use the periodic fundamental solutions, which was presented by Hasimoto [8] and is given by a Fourier series. It is expected to construct a method of fundamental solutions using periodic fundamental solutions given by the theta functions or elliptic functions [10, 5] as in this paper.

## References

- [1] (1994) A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. J. Comput. Appl. Math. 53 (3), pp. 353–370. External Links: Document Cited by: §1.
- [2] (1998) A charge simulation method for numerical conformal mapping onto circular and radial slit domains. SIAM J. Sci. Comput. 19 (4), pp. 1169–1187. External Links: Document Cited by: §1.
- [3] (2006) Elliptic functions. Cambridge University Press, Cambridge. External Links: Document Cited by: §2, footnote 3.
- [4] (2009) Exponential decay of errors of a fundamental solution method applied to a reduced wave problem in the exterior region of a disc. J. Comput. Appl. Math. 231, pp. 869–885. External Links: Document Cited by: §4.
- [5] (2018) Fast evaluation of the fundamental singularities of two-dimensional doubly periodic Stokes flow. J. Eng. Math. 111, pp. 95–110. External Links: Document Cited by: §4.
- [6] (1998) The method of fundamental solutions for elliptic boundary value problems. Adv. Comp. Math. 9, pp. 69–95. External Links: Document Cited by: §1.
- [7] (2004) Integral equation methods for stokes flow in doubly-periodic domains. J. Eng. Math. 48, pp. 157–170. External Links: Document Cited by: §1.
- [8] (1959) On the periodic fundamental solutions of the stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid. Mech. 5 (2), pp. 317–328. External Links: Document Cited by: §4.
- [9] (2008) Periodic fundamental solution of a two-dimensional Poisson equation. J. Phys. Soc. Japan 77 (10), pp. 104601. External Links: Document Cited by: footnote 4.
- [10] (2009) Periodic fundamental solution of the two-dimensional Stokes equations. J. Phys. Soc. Japan 78 (7), pp. 074401. External Links: Document Cited by: §4.
- [11] (1988) A mathematical study of the charge simulation method I. J. Fac. Sci. Univ. Tokyo, Sect IA 35 (3), pp. 507–518. Cited by: §1, §4.
- [12] (1990) Asymptotic error analysis of the charge simulation method in a jordan region with an analytic boundary. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 37, pp. 635–657. Cited by: §4.
- [13] (1978) Fluid transport by cilia between parallel plates. J. Fluid Mech. 86 (4), pp. 705–726. External Links: Document Cited by: §1.
- [14] (2011) Theoretical hydrodynamics. Dover, New York. Cited by: §1.
- [15] (1983) Charge simulation method and its applications. Morikita-Shuppan, Tokyo. Note: (in Japanese) Cited by: §1.
- [16] (1993) On “invariance” of schemes in the fundamental solution method. Trans. IPS Japan 34 (3), pp. 533–535. Note: (in Japanese) Cited by: footnote 2.
- [17] (1995) Comparison of conventional and “invariant” schemes of fundamental solutions method for annular domains. Japan J. Indust. Appl. Math. 12, pp. 61–85. Cited by: footnote 2.
- [18] (2003) A fundamental solution method for viscous flow problems with obstacles in a periodic array. J. Comput. Appl. Math. 152 (1–2), pp. 411–425. External Links: Document Cited by: §1, §4.
- [19] (2010) Fundamental solution method for two-dimensional stokes flow problems with one-dimensional periodicity. Japan J. Indust. Appl. Math. 27, pp. 191–215. External Links: Document Cited by: §1, §4.
- [20] (2011) A new theoretical error estimate of the method of fundamental solutions applied to reduced wave problems in the exterior region of a disk. J. Comput. Appl. Math. 235 (12), pp. 3395–3412. External Links: Document Cited by: §4.
- [21] (2014) Convergence of the invariant scheme of the method of fundamental solutions for two-dimensional potential problems in a jordan region. Japan J. Indust. Appl. Math. 31, pp. 231–262. External Links: Document Cited by: §4.
- [22] (2002) Numerical conformal mapping of periodic structure domains. Japan J. Indust. Appl. Math. 19, pp. 257–275. External Links: Document Cited by: §1.
- [23] (2008) Fundamental solution method for periodic plane elasticity. J. Numer. Anal. Indust. Appl. Math. (JNAIAM) 3 (3–4), pp. 249–267. Cited by: §1, §4.
- [24] (1979) Ground motion at canyons of arbitrary shape under incident sh waves. Int. J. Earthq. Eng. Struct. Dyn. 7, pp. 441–450. Cited by: §1.
- [25] (1974) A charge simulation method for the calculation of high voltage fields. IEEE Trans. Power Appar. Syst. PAS-93, pp. 1660–1668. External Links: Document Cited by: §1.
- [26] (1969) Note: dissertation, Tech. Univ. München Cited by: §1.
- [27] (1982) Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, pp. 13–26. External Links: Document Cited by: §1.