NCKU Math Colloquium / RCTS Seminar

DATE2015-12-03 16:10-17:00



TITLENumerical Investigations of Error in Generalized Finite Difference Method for Second-order Partial Differential Equations

ABSTRACT Abstract: In this study, we numerically investigated the convergence rate of error in the generalized finite difference method (GFDM) for second-order partial differential equation by adopting different-orders Taylor series. The GFDM is a newly-developed promising domain-type meshless method and only scattered nodes are required for numerical simulation. To use the moving least-square method in the GFDM can derive the expressions of spatial derivatives at every node and then the numerical solutions can be acquired via a collocation approach. The numerical procedures of the GFDM are very simple and the simulation for second-order partial differential equation is very stable. Theoretically speaking, the truncation errors due to the truncated terms of the Taylor series will principally dominate the numerical errors of the GFDM. In this study, we used second-order, third-order and fourth-order Taylor series in the moving least-squares method to derive the expressions of spatial derivatives, respectively. It can be expected that the numerical errors of the GFDM will decrease while either more nodes or higher-order Taylor series are used. Several numerical examples are provided to show the convergence rate of errors while different numbers of total nodes and different orders of Taylor series are adopted. The numerical estimations of convergence rate are very similar with the theoretical predictions in these examples. When the Neumann boundary condition is included, the numerical estimations of error tendency are slightly different from the theoretical predictions. From the provided comparisons, it can be concluded that the convergence rate of error in the GFDM by adopting different-orders Taylor series can be predicted, which would be very helpful to future engineering applications of the GFDM.