A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows

Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems

“…From the numerical results given in [36], it was found that the upwind scheme does not work well at nodes near the boundary layer when the perturbation parameter is not small. Recently, these upwind techniques have been applied to solve convection-dominated problems [37,38]. For solving hyperbolic problems, the upwind technique is also applicable [39] to get numerical solutions without non-physical oscillations.…”

Inverse differential quadrature method: mathematical formulation and error analysis

“…For example, in the field of structural mechanics, the DQM approach has been explored for static analysis [30][31][32], free vibration analysis [33][34][35], buckling analysis [29,36] and large deflection post-buckling analysis [37]. In the context of fluid mechanics applications, DQM has been widely applied to obtain solutions of convection problems [38][39][40] and Navier-Stokes equations [41,42], heat transfer analysis [43,44], and magnetohydrodynamic duct flow problems [45]. It is instructive to note that in [38][39][40]42] the so-called localized DQM is adopted in combination with RBF since this strategy allows versatility in using meshes or meshless systems [46].…”

“…In the context of fluid mechanics applications, DQM has been widely applied to obtain solutions of convection problems [38][39][40] and Navier-Stokes equations [41,42], heat transfer analysis [43,44], and magnetohydrodynamic duct flow problems [45]. It is instructive to note that in [38][39][40]42] the so-called localized DQM is adopted in combination with RBF since this strategy allows versatility in using meshes or meshless systems [46]. Financial engineering is another area where the merits of DQM have been explored for computational analysis [47].…”

“…On the other hand, for applications like wave propagation in space where uniformly distributed grid points are required for accurate numerical estimation, the localized DQM (LDQM), which applies DQM approximations within a small neighbourhood of the point of interest, is crucial to keep the balance between the accuracy and stability of the numerical estimates [25]. In recent times, remarkably in computational fluid mechanics applications, the so-called RBF-based DQ method, as well as the LDQM variant, are increasingly being adopted to extend the merits of the DQM, as already noted in [38][39][40]42]. For the purpose of the current work, the PDQ-based approach with a non-uniform grid structure is adopted.…”

Comparing RBF‐FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials

“…Since then, RBF‐FDs have been applied in many research areas: fluid dynamics (Shu et al, Ding et al, Shu et al, Shu et al, Ding et al, and Chan et al), heat transfer (Qajarjazi et al, Soleimani et al, and Martin and Fornberg), electrostatic potential and temperature by the Joule effect (Bararnia et al), vibration (Wu et al), and seismic modeling (Martin and Fornberg), among others.…”