In this paper, the Fragile Points Method (FPM) has been extended to solve the two-dimensional hyperbolic telegraph equation with specified initial and boundary conditions. Based on a naturally partitioned domain with scattered nodes and the Voronoi Diagram, the discretized FPM equations are derived by a Galerkin weak-form in the spatial domain and a finite difference scheme in the time domain. For the spatial discretization, discontinuous point-based polynomial trial and test functions are utilized, with numerical fluxes to ensure the consistency. For the time-domain discretization, theorems of unconditional stability and convergence for the telegraph equations are given. Numerical examples confirm the accuracy and robustness of the developed numerical method with uniform or random nodes and with different time increments. And as compared to several other meshless methods for the telegraph equation, it is shown that the developed FPM method has a better computational efficiency. This is because that a single-point quadrature rule is sufficient for evaluating the weak-form integral which gives a symmetric and sparse stiffness matrix, with the specifically-designed piecewise-linear trial and test functions.

在本文中，脆弱点法（FPM）已经扩展到求解具有指定初始和边界条件的二维双曲电报方程。基于具有分散节点的自然分区域和Voronoi图，离散化FPM方程由空间域的Galerkin弱形式和时域的有限差分格式导出。对于空间离散化，利用基于不连续点的多项式试验和测试函数，并通过数值通量来确保一致性。对于时域离散化，给出了电报方程的无条件稳定性和收敛性定理。数值例子证实了所开发的具有均匀或随机节点和不同时间增量的数值方法的准确性和鲁棒性。并且与其他几种电报方程的无网格方法相比，表明所开发的FPM方法具有更好的计算效率。这是因为单点求积规则足以使用专门设计的分段线性试验和测试函数来评估弱形式积分，该积分给出对称和稀疏的刚度矩阵。