Numerical Simulation of Partial Differential Equations via Local Meshless Method

Abstract

In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems.

Application of Least Squares Meshless Method in Multi-component High-Speed Non-equilibrium Reaction Jet

“…To solve this, Perrone and Kao [19]; Liszka and Orkisz [20] established the prototype of a meshless algorithm in the 1970s. The meshless algorithm [21][22][23] breaks through the limitations of grid division, and adopts the discrete nodes distributed in the boundary or within the computational domain, showing a huge advantage in analyzing the extremely large deformations and serious deformities of the multi-media, fluid-solid coupling, etc. Löhner et al [24] and Ortega et al [25], based on the finite point method, used Taylor-Galerkin format to successfully solve 2D and 3D compressible flows.…”

Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

“…Recently, a great effort has been expended to develop the exact and approximate behavior of fractional PDE. In this effort several enthusiastic methods have been applied for the solution of fractional PDE such as homotopy analysis method [9,10], Adomian decomposition method [11], fractional difference method [12], variational iteration method [13][14][15][16], homotophy perturbation method [17], Lie symmetry analysis [18], reproducing kernel method [19], Modified variational iteration method [20], meshless methods [21] and Chebychev spectral method [22].…”

Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

“…It is noted that the global meshless method (GMM), which is based on the global interpolation paradigm, has faced the problems of dense ill-conditioned matrices and finding the optimum value of the shape parameter. To avoid the limitations of the GMM, a local meshless method, which is based on local interpolation in the sub-domains, is used as a substitute to get a stable and accurate solution for the PDE models (see [31][32][33][34][35][36][37]).…”