Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T12:01:56.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

Published online by Cambridge University Press:  27 March 2015

C. S. Chen ,
Xinrong Jiang ,
Wen Chen  and
Guangming Yao
C. S. Chen
Affiliation:
Department of Engineering Mechanics, Hohai University, Nanjing, China Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, USA
Xinrong Jiang
Affiliation:
Bank of Nanjing, Nanjing, China, 210008
Wen Chen*
Affiliation:
Department of Engineering Mechanics, Hohai University, Nanjing, China
Guangming Yao
Affiliation:
Department of Mathematics, Clarkson University, Potsdam, NY, USA
*
*Corresponding author. Email addresses: [email protected] (C. S. Chen), [email protected] (X. R. Jiang), [email protected] (W. Chen), [email protected] (G. Yao)
Get access

Abstract

The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

Keywords

65N1065N30Method of fundamental solutionsmethod of particular solutionsboundary meshless methodpolyharmonic splinesradial basis functionsmodified Helmholtz equation
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 867 - 886
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Partridge, P., Brebbia, C., and Wrobel, L., The Dual Reciprocity Boundary Element Method, CMP/Elsevier, 1992.Google Scholar
[2]Nowak, A. and Neves, A., The multiple reciprocity boundary element method, Computational Mechanics Publications, 1994.Google Scholar
[3]Kupradze, V. and Aleksidze, M., The method of functional equations for the approximate solution of certain boundary value problems, U.S.S.R. Computational Mathematics and Mathematical Physics, 4 (1964), 82–126.CrossRefGoogle Scholar
[4]Golberg, M. and Chen, C., The method of fundamental solutions for potential, Helmholtz and diffusion problems, in: Golberg, M. (Ed.), Boundary Integral Methods: Numerical and Mathematical Aspects, WIT Press, (1998), 103–176.Google Scholar
[5]Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), 389–396.Google Scholar
[6]Chen, C., Brebbia, C. and Power, H., Dual reciprocity method using compactly supported radial basis functions, Comm. Num. Meth. Eng., 15 (1999), 137–150.Google Scholar
[7]Cho, H., Golberg, M., Muleshkov, A. and Li, X., Trefftz methods for time dependent partial differential equations, Computers, Materials, and Continua, 1 (2004), 1–38.Google Scholar
[8]Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22 (1985), 644–669.CrossRefGoogle Scholar
[9]Fairweather, G. and Karageorghis, A., The method of fundamental solution for elliptic boundary value problems, Advances in Computational Mathematics, 9 (1998), 69–95.Google Scholar
[10]Cheng, R., Delta-trigonometric and Spline Methods using the Single-layer Potential Representation, Ph.D. thesis, University of Maryland (1987).Google Scholar
[11]Katsurada, M. and Okamoto, H., A mathematical study of the charge simulation method, Journal of the Faculty of Science, University of Tokyo, Section 1A, 35 (1988), 507–518.Google Scholar
[12]Katsurada, M., Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary, Journal of the Faculty of Science of Tokyo University, Section 1A, 37 (1990), 635–657.Google Scholar
[13]Katsurada, M., Charge simulation method using exterior mapping functions, Japan Journal of Industrial and Applied Mathematics, 11 (1994), 47–61.CrossRefGoogle Scholar
[14]Katsurada, M. and Okamoto, H., The collocation points of the fundamental solution method for the potential problem, Computers and Mathematics with Applications, 31 (1996), 123–137.CrossRefGoogle Scholar
[15]Li, M., Chen, C.S. and Karageorghis, A., The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions, Computers and Mathematics with Applications, 66 (2013), 2400–2424.CrossRefGoogle Scholar
[16]Muleshkov, A., Golberg, M. and Chen, C.S., Particular solutions of Helmholtz-type operators using higher order polyharmonic splines, Comp. Mech. 23 (1999) 411–419.CrossRefGoogle Scholar
[17]Muleshkov, A., Chen, C.S, Golberg, M. and Cheng, A.-D., Analytic particular solutions for inhomogeneous Helmholtz-type equations, in: Atluri, S., Brust, F. (Eds.), Advances in Computational Engineering & Sciences, Tech Science Press, (2000) 27–32.Google Scholar
[18]Golberg, M., Muleshkov, A., Chen, C.S. and Cheng, A.-D., Polynomial particular solutions for certain kind of partial differential operators, Numerical Methods for Partial Differential Equations, 19 (2003), 112–133.CrossRefGoogle Scholar
[19]Yao, G., Chen, C.S., Kolibal, J., A localized approach for the method of approximate particular solutions, Computers and Mathematics with Applications, 61 (2011), 545–559.Google Scholar