The Fragile Points Method (FPM) to solve two-dimensional hyperbolic telegraph equation using point stiffness matrices

Abstract

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.