Journal of Scientific Computing

, Volume 78, Issue 3, pp 1632–1658 | Cite as

A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation

  • Yaning Xie
  • Wenjun YingEmail author


Based on the kernel-free boundary integral method proposed by Ying and Henriquez (J Comput Phys 227(2):1046–1074, 2007), which is a second-order accurate method for general elliptic partial differential equations, this work develops it to be a fourth-order accurate version for the modified Helmholtz equation. The updated method is in line with the original one. Unlike the traditional boundary integral method, it does not need to know any analytical expression of the fundamental solution or Green’s function in evaluation of boundary or volume integrals. Boundary value problems under consideration are reformulated into Fredholm boundary integral equations of the second kind, whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method, the Richardson iteration. During each iteration, a Cartesian grid based nine-point compact difference scheme is used to discretize the simple interface problem whose solution is the boundary or volume integral in the BIEs. The resulting linear system is solved by a fast Fourier transform based solver, whose computational work is roughly proportional to the number of grid nodes in the Cartesian grid used. As the discrete boundary integral equations are well-conditioned, the iteration converges within an essentially fixed number of steps, independent of the mesh parameter. Numerical results are presented to verify the solution accuracy and demonstrate the algorithm efficiency.


Elliptic partial differential equation Kernel-free boundary integral method Cartesian grid method Nine-point compact difference scheme 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018
Corrected publication October/2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiao Tong UniversityMinhang, ShanghaiPeople’s Republic of China
  2. 2.School of Mathematical Sciences, MOE-LSC and Institute of Natural SciencesShanghai Jiao Tong UniversityMinhang, ShanghaiPeople’s Republic of China

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