On a high-order Gaussian radial basis function generated Hermite finite difference method and its application

Recently, several improvements over the radial basis function generated finite difference method have been appeared numerically and theoretically in literature. Suitable convergence orders can be obtained once they are combined with stable evaluation schemes. However, some numerical issues remain such as sensitivity to the node layout, and equal order of convergence to the FD-type methods having the same stencil. Here, we propose the weights of the radial basis function generated Hermite finite difference formulation employing the Gaussian kernel on graded meshes. The theoretical rates reveal higher convergence orders for approximating function derivatives. Application of this methodology for solving two-dimensional time-dependent Heston partial differential equation is furnished in detail. Finally, numerical tests confirm the theoretical estimates.