The enhanced boundary knot method with fictitious sources for solving Helmholtz-type equations

Abstract

The boundary knot method (BKM) is a boundary meshless method for solving homogeneous boundary value problems. The fact that the BKM uses non-singular general solution without the need of source points outside the domain is at the expense of the accuracy when compared to the well-known method of fundamental solutions. In this paper, a new approach called the enhanced BKM (EBKM) is proposed to greatly improve the accuracy of the BKM by introducing fictitious sources which can be placed inside and/or outside the domain quite freely. We also adopt various techniques to place these fictitious sources in an optimal way.

Keywords:

• Boundary knot method
• Hemholtz-type equations
• method of fundamental solutions
• fictitious sources technique

AMS SUBJECT CLASSIFICATION:

• 65N20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

We provide the following MATLAB${}^{\circledR }$ code for the solution of Example 1 using LOOCV. For the case using effective condition number, Lines 45–47 can be replaced by Lines 48–51. The following code can be run using MATLAB${}^{\circledR }$ version R2021a or above.

Additional information

Funding

The author is grateful for the support of the grants from Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515110680), Guangzhou Basic and Applied Basic Research (No. 202102020340), Opening Project of Guangdong Province Key Laboratory of Computational Science (No. 2021023) and Youth Science and Technology Research Foundation of Shanxi Province (No. 202103021223059), School Foundation of Taiyuan University of Technology (No. 2022QN099).