Adaptive Numerical Integration Based on Local Taylor Expansions

This study presents a numerical integration method based on local Taylor polynomial approximations over extended domains. By allowing the expansion point to vary along the interval, x0=x0,i, accurate local representations are obtained with few terms. The method can be interpreted as a cell-based, piecewise quadrature scheme with error control provided by the Taylor remainder. It avoids the limitations of global Taylor expansion and remains efficient for large domains. The approach is extended to two-dimensional problems over regular and irregular regions, achieving high accuracy with low computational cost compared to classical methods.