Mathematics

CLC number: O242

On Numerical Examples of Boundary Knot Method for Helmholtz-Type Equation

边界节点法求解Helmholtz型方程的数值算例研究

General Education Center, Zhengzhou Business University, Zhengzhou 451200, Henan, China

Received: 15 December 2024

Abstract

The boundary knot method (BKM) is a simple boundary-type meshless method. Due to the use of non-singular general solutions rather than singular fundamental solutions, BKM does not need to consider the artificial boundary. Therefore, this method has the merits of purely meshless, easy to program, high solution accuracy and so on. In this paper, we investigate the effectiveness of the BKM for solving Helmholtz-type problems under various conditions through a series of novel numerical experiments. The results demonstrate that the BKM is efficient and achieves high computational accuracy for problems with smooth or continuous boundary conditions. However, when applied to discontinuous boundary problems, the method exhibits significant numerical instability, potentially leading to substantial deviations in the computed results. Finally, three potential improvement strategies are proposed to mitigate this limitation.

摘要

边界节点法是一种简单的边界型无网格方法。由于采用非奇异通解而不是奇异基本解，无需考虑人工边界的问题，因此边界节点法具有纯无网格、易于编程、求解精度高等优点。本文通过一系列新的数值实验，研究了边界节点法在不同条件下求解Helmholtz型问题的有效性。研究结果表明，对于具有光滑或连续边界条件的问题，边界节点法有效且具有较高的计算精度。然而，在处理不连续边界问题时，该方法存在明显的数值不稳定性，计算结果可能出现显著偏差。最后，针对这一局限性，提出了三个可能的改进方向。