A new approach to design high-order schemes

In this paper a new approach, MPDE-aided method of undetermined coefficient, is proposed to design high-order finite difference schemes. This designing approach differs from other approaches in that it is constructed on the analysis theory of the modified partial differential equation. By this approach, not only the deduction of different schemes is unified, but also some multi-level highly accurate schemes are developed. Numerical tests for these high-order schemes are presented to verify their quality.