Finite-difference stable stencils based on least-square quadric fitting

In this paper a procedure to obtain new finite difference stencils is given. We investigate the particular branch of cases where the order of the finite difference stencil is lower than the amount of grid points. New finite difference stencils are obtained and compared with traditional ones from literature. General properties and advantages of the proposed stencils are investigated, in particular the improvements obtained in stability when solving ill-posed problems.