Implementation of generalized finite difference formulas and truncation errors for each derivative

Abstract

Through Taylor series expansion, a generalized formulation which can show any finite difference expression along with its truncation error for each derivative has been introduced in the present study. Thereafter, several well-known finite difference and finite element formulas for the first and second derivatives are demonstrated to compare with their truncation errors. As to the applications, higher-order terms of truncation error can be incorporated into finite difference equation to get more accurate results. In addition, a new scheme is proposed to generate an optimum finite difference equation where its total truncation error is minimized.