## 1 Introduction

The method of fundamental solutions, or the charge simulation method [5, 12], is a fast solver of potential problems

where is a domain in the -dimensional Euclidean space
and is a function given on the boundary .
In the two-dimensional problems ,
equalizing the Euclidean plane with the complex plane ,
the method approximates the solution by
^{2}^{2}2
The approximation shown here is that of the invariant scheme proposed by Murota
[13, 14].

(1) |

where are points given in and are unknown real constants such that

(2) |

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 following collocation condition on .

(3) |

where are points given on . We call the “collocation points”. The equation (3) is rewritten as

(4) |

The equations (2) and (4) form a system of linear equations for . We obtain by solving the linear system (2) and (4), and we 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 achieves high accuracy such as exponential convergence under some conditions [9, 10, 19]. It was first used for electrostatic problems [25, 26], and now it is used in widely in science and engineering such as Helmholtz equation problems [4, 18] and studies on scattering of earthquake wave [24] and so on.

The method of fundamental solutions is also applied to the approximation of complex analytic functions. Let be an complex analytic function in some complex domain which is to be approximated and satisfies some boundary condition. Since the real part is a harmonic function in , it can be approximated using the right hand side of (1), and the imaginary part , the conjugate harmonic function of , is approximated using

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

(5) |

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

In this paper, we examine the problem of two-dimensional potential flow past a doubly-periodic array of obstacles as shown in Figure 1. A two-dimensional potential flow in a domain is characterized by a complex velocity potential , a complex analytic function in which gives the velocity field by and satisfies the boundary condition

(6) |

Physically, the condition (6) means that the fluid flows along the boundary since the contour lines of give the streamlines. Therefore, we can obtain the complex velocity potential by approximating it by the form (5). However, it is difficult to apply the method of fundamental solutions to our problem because the solution of our problem obviously involves a doubly-periodic function due to the periodicity of the problem, which it is difficult to approximate by the form (5) of the conventional method. To overcome this challenge, we propose a new method of fundamental solution method for our problem. In the proposed method, we approximate the solution involving a doubly-periodic function using a linear combination of periodic fundamental solutions, that is, complex logarithmic potentials with sources in a doubly-periodic array which is constructed by the Weierstrass sigma functions. Our method inherits the advantages of the conventional method of fundamental solutions and give an approximate solutions with the periodicity which we want. The author proposed a method of fundamental solution for the problem of two-dimensional potential flow with double periodicity, where we used the periodic fundamental solution constructed by the theta functions [23].

The author has presented many works on methods of fundamental solution for periodic problems. In [20], we presented a method of fundamental solutions for numerical conformal mappings of singly-periodic complex domains, where we approximated the mapping function using singly-periodic fundamental solutions, that is, the complex logarithmic potentials with sources in a singly-periodic array. The author proposed methods of fundamental solutions for the problems of Stokes flow past a periodic array of obstacles [15, 21, 16, 17], where the solutions are approximated using the periodic fundamental solutions of the Stokes equation, that is, the Stokes flow induced by concentrated forces in a periodic array. As to works on periodic Stokes flow, Zick and Homsy [27] proposed an integral equation method for three-dimensional Stokes flow problems with a three-dimensional periodic array of spheres. Greengard and Kropinski [6] proposed an integral equation method for two-dimensional Stokes flow problems in doubly-periodic, where the approximate solution is given as a complex variable formulation and the fast multipole method is used. Liron [11] proposed a study on Stokes flow due to infinite arrays of Stokeslets and its application to fluid transport by cilia. In addition, the author proposed a method of fundamental solutions for periodic plane elasticity [22], 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 gives our method of fundamental solutions for periodic-potential flow. Section 3 presents some numerical examples which show the effectiveness of our method. In Section 4, we conclude this paper and presents some problems related to future studies.

## 2 Method of fundamental solutions

We examine the problem of two-dimensional potential flow past a doubly-periodic array of obstacles. In terms of mathematics, the flow domain is given by

where is one of obstacles, in terms of mathematics, a simply-connected domain in , and , is given by

with the complex number such that giving the periods of our problem.

The complex velocity potential of our problem is a complex analytic function in the domain satisfying the boundary condition

(7) |

Additionally, we assume that the average of the flow is the uniform flow, in whose direction we take the real axis, when it is observed along the direction of , that is,

(8) |

where is the magnitude of the uniform flow, is the curve in connecting two points and

is the unit normal vector on

. Using , the condition (8) is rewritten as(9) |

Therefore, our problem is to find a complex velocity potential , an analytic function in such that it satisfies the boundary condition (7) and the condition (9).

We propose to approximate the complex velocity potential in the following form according to [8].

(10) |

where are points given in , are unknown real coefficients such that

(11) |

is the Weierstrass sigma function [3], and is given by using the Weierstrass zeta function . We call the real coefficients the “charges” and the “charge points”. Since is an entire function with simple zeros at , , the functions appearing on the right hand side of (10) are complex logarithmic potential with sources in a doubly-periodic array. The approximate potential satisfies the condition (9). In fact, we have

where we used the pseudo-periodicity of the sigma function

(12) |

with , and (11). The approximate potential satisfies the pseudo-periodicity

(13) |

due to (12) and (11). Then, the complex velocity satisfies

that is, it is an elliptic function with periods and . Regarding the boundary condition (7), we pose the following collocation condition on

(14) |

where are points given on , and is a real constant. We call the “collocation condition”. The collocation condition (14) is rewritten as

(15) |

which form a system of linear equations for and together with (11). We obtain the charges by solving the linear system (11) and (15) and obtain the approximate potential . By (14), approximately satisfies the boundary condition (7) on the boundaries of other obstacles , due to the pseudo-periodicity (13).

## 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.

We computed the two-dimensional potential flow past a doubly-periodic array of cylinders of radius

where | |||

and are the periods of the array such that by our method. In our method, we took the charge points and the collocation points respectively as

(16) |

where is a constant such that , which is taken as in the example. Figure 2 shows the streamline of the flows for some pairs of

x y | x y |

x y | x y |

To estimate the accuracy of our method, we computed

where is the constant appearing in (15). The value shows how accurately the approximate potential satisfies the boundary condition (7). Figure 3 shows the value computed for the example with the parameter in (16) taken as some values. The figures show that decays exponentially as the number of unknowns increases. Table 1 shows the decay rates of computed by the least square fitting using the fit command of the software gnuplot. The table shows that the decay rate of roughly obeys the rule

n e | n e |

n e | n e |

## 4 Concluding Remarks

In this paper, we examined the problems of two-dimensional potential flow past a doubly-periodic array of obstacles and proposed a method of fundamental solutions for these problems. It is difficult to apply the conventional method of fundamental solution to our problems because the solution involves a doubly-periodic functions. We proposed a method of fundamental solutions for our problems, where the solution is approximated using the periodic fundamental solutions, that is, the complex logarithmic potentials with sources in a doubly-periodic array and constructed by the Weierstrass sigma functions. The proposed method inherits the advantages of the conventional method and approximates well the solution involving a periodic function. The numerical examples showed the effectiveness of our method.

We have two issues regarding this paper for future studies. The first problem is to extend our method to other periodic problems than the Laplace equation such as the Stokes equation. In the author’s previous works on periodic Stokes flow [15, 21, 16], the solutions are approximated using the periodic fundamental solutions which was presented by Hasimoto [7] and expressed by a Fourier series. In these years, Hasimoto presented the periodic fundamental solutions of the Stokes equation using the Weierstrass elliptic functions [8], and it is interesting to construct a method of fundamental solutions using these periodic fundamental solutions.

The second problem is a theoretical study on the accuracy of our method. Theoretical error estimates on method of fundamental solutions are presented for special problems such as two-dimensional potential or Helmholtz equation problems in a disk [9, 4, 18] and two-dimensional potential problems in a domain with an analytic boundary [10, 19]; however, there still remain many problems including the problem of this paper to which theoretical error estimates are to be given. This is one of the most important problems on the method of fundamental solutions.

## 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.
- [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: §1, §4.
- [5] (1998) The method of fundamental solutions for elliptic boundary value problems. Adv. Comp. Math. 9, pp. 69–95. External Links: Document Cited by: §1.
- [6] (2004) Integral equation methods for stokes flow in doubly-periodic domains. J. Eng. Math. 48, pp. 157–170. External Links: Document Cited by: §1.
- [7] (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.
- [8] (2008) Periodic fundamental solution of a two-dimensional Poisson equation. J. Phys. Soc. Japan 77 (10), pp. 104601. External Links: Document Cited by: §2, §4.
- [9] (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.
- [10] (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: §1, §4.
- [11] (1978) Fluid transport by cilia between parallel plates. J. Fluid Mech. 86 (4), pp. 705–726. External Links: Document Cited by: §1.
- [12] (1983) Charge simulation method and its applications. Morikita-Shuppan, Tokyo. Note: (in Japanese) Cited by: §1.
- [13] (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.
- [14] (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.
- [15] (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.
- [16] (2006) A fundamental solution method for three-dimenstional viscous flow problems with obstacles in a periodic array. J. Comput. Appl. Math. 193, pp. 302–318. External Links: Document Cited by: §1, §4.
- [17] (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.
- [18] (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: §1, §4.
- [19] (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: §1, §4.
- [20] (2002) Numerical conformal mapping of periodic structure domains. Japan J. Indust. Appl. Math. 19, pp. 257–275. External Links: Document Cited by: §1.
- [21] (2006) A fundamental solution method for three-dimensional stokes flow problems with obstacles in a planar periodic array. J. Comput. Appl. Math. 189, pp. 622–634. External Links: Document Cited by: §1, §4.
- [22] (2008) Fundamental solution method for periodic plane elasticity. J. Numer. Anal. Indust. Appl. Math. (JNAIAM) 3 (3–4), pp. 249–267. Cited by: §1.
- [23] (2020) Method of fundamental solutions for the problem of doubly-periodic potential flow. Note: arXiv:2006.12763 Cited by: §1.
- [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.