TY - JOUR
T1 - A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels
AU - Mishra, Pankaj K.
AU - Fasshauer, Gregory E.
AU - Sen, Mrinal K.
AU - Ling, Leevan
N1 - Funding Information:
This work was partially supported by a Hong Kong Research Grant Council GRF Grant.
PY - 2019/5/1
Y1 - 2019/5/1
N2 - Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.
AB - Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.
KW - Ill-conditioning
KW - Partial differential equations
KW - Radial basis functions
KW - RBF-FD
UR - http://www.scopus.com/inward/record.url?scp=85059168095&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2018.12.027
DO - 10.1016/j.camwa.2018.12.027
M3 - Journal article
AN - SCOPUS:85059168095
SN - 0898-1221
VL - 77
SP - 2354
EP - 2368
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
IS - 9
ER -