Approximation error of the Lagrange reconstructing polynomial

Abstract

The reconstruction approach [C.W. Shu, High-order weno schemes for convection-dominated problems, SIAM Rev. 51 (1) (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 − 1 2 Δ x , x + 1 2 Δ x ] are equal to f ( x ) (assuming a homogeneous grid of cell-size Δ x ). We study the deconvolution problem [A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes III, J. Comput. Phys. 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 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.