A note on the direct approximation of derivatives in rational radial basis functions partition of unity method and its application to the convection-diffusion equations

This paper proposes a direct rational radial basis functions partition of unity (D-RRBF-PU) approach to compute derivatives of functions with steep gradients or discontinuities. The novelty of the method concerns how derivatives are approximated. More precisely, all derivatives of the partition of unity weight functions are eliminated while we compute the derivatives of the local rational approximants in each patch. As a result, approximate derivatives are obtained more easily and quickly than those obtained in the standard formulation. The corresponding error bounds are briefly discussed. Some numerical results are presented to show the technique’s potential. Further, a comparison between the proposed method and the standard RRBF-PU approximation is conducted in terms of the accuracy and the used CPU time. As an application, we develop this meshfree approximation combined with an explicit fourth-order Runge-Kutta time discretization to find the numerical solution of the convection-diffusion equations in two dimensions.