Quasi-a priori truncation error estimation and higher order extrapolation for non-linear partial differential equations

In this paper, we show how to accurately estimate the local truncation error of partial differential equations in a quasi-a priori way. We approximate the spatial truncation error using the τ-estimation procedure, which aims to compare the discretisation on a sequence of grids with different spacing. While most of the works in the literature focused on an a posteriori estimation, the following work develops an estimator for non-converged solutions. First, we focus the analysis on one- and two-dimensional scalar non-linear test cases to examine the accuracy of the approach using a finite difference discretisation. Then, we extend the analysis to a two-dimensional vectorial problem: the Euler equations discretised using a finite volume vertex-based approach. Finally, we propose to analyse a direct application: τ-extrapolation based on non-converged τ-estimation. We demonstrate that a solution with an improved accuracy can be obtained from a non-a posteriori error estimation approach.