ABSTRACT
Recent developments in a higher-order approximation to both classical finite difference (FD) and meshless finite difference methods (MFDM [1]) are discussed, as well as validation of this approach through the analysis of 1D boundary value problems. The higher-order approximation concept has been introduced in [2] and developed further in [3]. It is based on expansion of the FD operators into Taylor series. The same mesh, as in the case of the lower-order approximation, is used, but selected higher-order terms are included. For boundary value problems of the m-th order, the approach provides results that do not depend on the quality of the FD operator used. They are exact within the 2n-th order Taylor series. In the present study not only smooth solutions, but also jumps of a searched function and its derivatives may be accounted for. Preliminary 1D tests done so far provided very encouraging results.