Convergence properties of the radial basis function-finite difference method on specific stencils with applications in solving partial differential equations

We consider the problem of approximating a linear differential operator on several specific stencils using the radial basis function method in the finite difference scheme. We prove a linear convergence order on a non-equispaced five-point stencil. Then, we discuss how the convergence rate can be boosted up to the second- order on an equispaced stencil. Moreover, we show that including additional nearby nodes (six to twelve) in the stencil does not improve the convergence rate, thus increasing the computational load without enhancing convergence. To overcome this limitation, we propose a stencil that accelerates the convergence up to four using a nine-point stencil, unlike existing approaches which are based on thirteen-point equispaced stencils to achieve such an order of convergence. To support our findings, we conduct numerical experiments by solving Poisson equations and a parabolic problem.