Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations - Shu, C; Ding, H; Yeo, K.S (Computer methods in applied mechanics and engineering, 2003)
Local radial basis function-based differential quadrature method is presented in detail in this paper. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with “truly” mesh-free property. In this paper, we mainly concentrate on the multiquadric RBFs since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to the simulation of natural convection in a square cavity. Excellent numerical results are obtained on an irregular knot distribution.