Approximation error of the Lagrange reconstructing polynomial
This provides a theoretical analysis of the error in Lagrange-based reconstruction for numerical differentiation, which is of interest to researchers in numerical analysis and computational fluid dynamics.
The paper studies the deconvolution problem linking the Taylor polynomials of the dual function and the original function in the reconstruction approach for numerical differentiation. It obtains explicit solutions via rational numbers defined by a recurrence or generating function, and applies these to determine the approximation error of the Lagrange reconstructing polynomial on homogeneous grids.
The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}Δx,x+\tfrac{1}{2}Δx]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $Δx$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $τ_n$ defined by a recurrence relation, or determined by their generating function, $g_τ(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.