Superconvergence of Centered Finite Difference Approximations
Abstract
The finite difference (FD) method is commonly used to approximate derivatives of smooth functions, with accuracy typically determined by stencil size and derivative order. However, certain centered stencils exhibit unexpectedly higher accuracy, a phenomenon known as superconvergence, which has been observed in practice but lacks rigorous explanation. We present a mathematical framework for superconvergence in centered FD approximations based on Taylor expansions of the truncation error and the resulting linear system for the FD coefficients. By analyzing symmetry properties of these coefficients and their interaction with the parity of the derivative order, we identify conditions under which higher-order error terms cancel. We show that superconvergence occurs for odd-order derivatives with even centered stencils and for even-order derivatives with odd centered stencils, while no superconvergence occurs for even derivatives with even centered stencils. Numerical experiments in MATLAB confirm the predicted convergence rates.
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BibTeX
@misc{bencomo2026superconvergence,
title = {{Superconvergence of Centered Finite Difference Approximations}},
author = {Mario J. Bencomo and Joseph Igot and Emma Maltes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15831},
eprint = {2609.15831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15831},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Superconvergence of Centered Finite Difference Approximations
AU - Mario J. Bencomo
AU - Joseph Igot
AU - Emma Maltes
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15831
UR - https://arxiv.org/abs/2609.15831
AB - The finite difference (FD) method is commonly used to approximate derivatives of smooth functions, with accuracy typically determined by stencil size and derivative order. However, certain centered stencils exhibit unexpectedly higher accuracy, a phenomenon known as superconvergence, which has been observed in practice but lacks rigorous explanation. We present a mathematical framework for superconvergence in centered FD approximations based on Taylor expansions of the truncation error and the resulting linear system for the FD coefficients. By analyzing symmetry properties of these coefficients and their interaction with the parity of the derivative order, we identify conditions under which higher-order error terms cancel. We show that superconvergence occurs for odd-order derivatives with even centered stencils and for even-order derivatives with odd centered stencils, while no superconvergence occurs for even derivatives with even centered stencils. Numerical experiments in MATLAB confirm the predicted convergence rates.
ER -