Global and compact meshless schemes for the unsteady convection-diffusion equation

The numerical solution of convection-diffusion equation has been a long standing problem and many numerical schemes which attempt to find stable and accurate solutions for convection dominated cases have to resort to artificial dissipation to stabilize the numerical solution. In this paper, we investigate the application of global and compact meshless collocation techniques with radial basis functions for solving the unsteady convection-diffusion equation. We employ the method of lines approach to discretize the governing operator equation. The stability of both explicit and implicit time stepping schemes are analyzed. Numerical results are presented for one-dimensional and two-dimensional problems using various globally supported radial basis functions such as multiquadric (MQ), inverse multiquadric (IMQ), Gaussian, thin plate splines (TPS) and quintics. Numerical studies suggest the global MQ, IMQ and Guassian (when the shape parameter is prperly tuned) have very high convergence rate than TPS and appears that the global meshless collocation techniques require a very dense set of collocation points in order to achieve accurate results for high Peclet numbers. For the compact supported RBFs, it is found that as the support parameter is increased, the sparsity decreases resulting in a better accuracy but at additional computational cost