Stability of Time-Marching MPS–MFS for Wave Equations

The two-stage meshless method utilizing the method of particular and fundamental solutions two-stage MPS–MFS was originally proposed to solve steady-state boundary value problems. Despite its great success in handling complicated irregular domains, when applied as a spatial discretization technique to solve time-dependent problems, especially for hyperbolic PDEs, the two-stage MPS–MFS seems to lead to numerical stability issues for conventional time-marching schemes. In addition to the unstable phenomena observed in some existing works, a concrete example on wave equation in this paper also shows the instability of this methodology, even though the utilized time-marching schemes are unconditionally stable. To investigate and explain this instability, we intend to conduct a thorough analysis of the time-marching MPS–MFS method for wave equations. To our best knowledge, this is the first time that the stability issue of the MPS–MFS methodology for hyperbolic PDEs is studied. To compensate for the instability caused by the two-stage MPS–MFS, we also propose a novel time-marching meshless method by combining the method of polynomial particular solutions (MPPS) and the Rothe’s method. This innovative method has a distinct solution mechanism and outperforms the time-marching MPS–MFS method in terms of stability and accuracy. Numerical results are provided to support our findings.