On the relation of truncation and approximation errors for the set of solutions obtained by different numerical methods

TL;DR

This paper investigates the relationship between truncation and approximation errors in numerical solutions of 2D steady inviscid compressible flow, proposing methods to estimate errors and analyze their independence and effects.

Contribution

It introduces a novel analysis of the correlation between truncation and approximation errors using geometric angles and compares two error estimation procedures in compressible flow simulations.

Findings

Angles between approximation errors are far from zero, enabling error norm estimation.

The comparison of two error estimation procedures shows acceptable effectivity indices.

Results include error analysis for supersonic flows with shock waves.

Abstract

The truncation and approximation errors for the set of numerical solutions computed by methods based on the algorithms of different structure are calculated and analyzed for the case of the two-dimensional steady inviscid compressible flow. The truncation errors are calculated using a special postprocessor, while the approximation errors are obtained by the comparison of the numerical solution and the analytic one. The extent of the independence of errors for the numerical solutions may be estimated via the Pearson correlation coefficient that may be geometrically expressed by the angle between errors. Due to this reason, the angles between the approximation errors are computed and related with the corresponding angles between the truncation errors. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The analysis of the distances between these solutions provides another approach to the estimation of the error. The comparison of the error norms, obtained by these two procedures, is provided that demonstrates the acceptable values of their effectivity indices. The results of the approximation error norm estimation for the supersonic flows, containing shock waves, are presented. The measure concentration phenomenon and the algorithmic randomness give some insights into these results.