An estimate of accuracy for interpolant numerical solutions of a PDE problem

TL;DR

This work presents an a priori error estimate for a piecewise cubic interpolant $V$ built from a coarse Euler-forward-in-time, centered-in-space solution of a nonlinear scalar conservation-law PDE, comparing it to a finer-grid solution $U$. By employing cubic Hermite-type interpolation on each coarse cell and enforcing a CFL-stability framework, the authors derive a concrete bound for the interpolant error that scales linearly with the coarse spatial step $h$, given a controllable initial-data condition. They further show that, under certain conditions (including linear flux and turbulence-like scenarios), the interpolant inherits stability properties and provides uniform spatial error control, offering a cost-effective alternative to solving on the finer grid. The results connect to classical finite-element error analyses and quantify the trade-off between accuracy and computational cost for interpolated, lower-resolution solutions.

Abstract

In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear differential problem. We suppose the numerical solution U is computed using a grid with a small linear step and interval time Tu, while the polynomial approximation V is an interpolation of the values of a numerical solution on a less fine grid and interval time Tv << Tu. The estimate shows that the interpolant solution V can be, under suitable hypotheses, a good approximation and in general its computational cost is much lower of the cost of the fine numerical solution. We present two possible applications to linear case and periodic case.

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Theorems & Definitions (18)

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