Simulation of linear and nonlinear advection-diffusion problems by the direct radial basis function collocation method

A simple direct radial basis function collocation method is proposed for the advection-diffusion equations. First, a new scheme of multiquadric radial basis function (RBF) is proposed. Then, the newly-proposed radial basis function can be directly used to deal with the time variable as well as space variables encountered in the advection-diffusion equations. This is realized by considering conventional time steps as collocation points. Numerical results for several advection-diffusion equations with different values of Pe´clet numbers show that the direct radial basis function collocation method performs well for both linear and nonlinear cases.