On the error propagation of semi-Lagrange and Fourier methods for advection problems

TL;DR

This paper investigates how errors propagate in various high-precision numerical schemes for advection problems, comparing Fourier, semi-Lagrangian, and discontinuous Galerkin methods through theoretical analysis and numerical experiments.

Contribution

It provides a detailed comparison of error propagation in Fourier, semi-Lagrangian, and discontinuous Galerkin methods, explaining their different behaviors and proposing modifications to reduce error growth.

Findings

Fourier-based methods are exact but affected by round-off errors leading to linear error growth.

Semi-Lagrangian methods' error propagation aligns with worst-case estimates, increasing proportionally with time steps.

Discontinuous Galerkin semi-Lagrangian methods do not exhibit the typical error increase, explained by their conservative properties.

Abstract

In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme.