Comparative analysis of the improved boundary knot and fundamental solutions methods for complex multi-connected Helmholtz-type equations

In this paper, the performance of the proposed improved boundary knot method and the method of fundamental solutions in solving Helmholtz-type equations within multi-connected domains is investigated. The method of fundamental solutions typically requires multiple layers of source points, resulting in a tedious and time-consuming process of optimizing their distribution. Although the traditional boundary knot method circumvents this challenge by using non-singular general solutions, it often struggles to deliver satisfactory accuracy for complex problems. To address these limitations, the improved boundary knot method that incorporates a ghost points technique is proposed. These ghost points can be positioned flexibly in various configurations, such as circular or cloud-like patterns, located either inside or outside the problem domain. To further optimize the ghost points' position, we study the influence of the free parameter ghost radius R, where two strategies, namely the effective condition number and economic effective condition number, are employed and analyzed. Finally, various examples demonstrate that the improved boundary knot method outperforms the conventional version. Compared to the method of fundamental solutions, it simplifies the placement of source/ghost nodes while maintaining accuracy. Code is available at https://github.com/LT306/one/tree/main/IBKM