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

Einkemmer L, Ostermann A - Comput Math Appl (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Innsbruck, Austria.

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.

No MeSH data available.


The  error of the Lagrange, spline, dG, and FFT based methods, as a function of the number of time steps, is shown. The FFT routine from the FFTW library is used. The polynomial degree is denoted by  and the number of cells/grid points is denoted by . As a reference two black lines of slope 1 are drawn.
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f000005: The error of the Lagrange, spline, dG, and FFT based methods, as a function of the number of time steps, is shown. The FFT routine from the FFTW library is used. The polynomial degree is denoted by and the number of cells/grid points is denoted by . As a reference two black lines of slope 1 are drawn.

Mentions: In Fig. 1, we compare the error propagation for a Lagrange interpolation, a discontinuous Galerkin method, and the Fourier approximation using the initial value (4)u(0,x)=12+cosπx on the interval with periodic boundary conditions (the same interval and periodic boundary conditions are used for all simulations in this paper). As expected, initially the FFT method achieves a performance close to machine precision. Note, however, that the error growth is linear in the number of time steps. However, from a stochastic description of the round-off error one would expect an error growth proportional to the square root in the number of time steps. Let us postpone the detailed investigation of this issue until Section  5.


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

Einkemmer L, Ostermann A - Comput Math Appl (2015)

The  error of the Lagrange, spline, dG, and FFT based methods, as a function of the number of time steps, is shown. The FFT routine from the FFTW library is used. The polynomial degree is denoted by  and the number of cells/grid points is denoted by . As a reference two black lines of slope 1 are drawn.
© Copyright Policy - CC BY
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4375614&req=5

f000005: The error of the Lagrange, spline, dG, and FFT based methods, as a function of the number of time steps, is shown. The FFT routine from the FFTW library is used. The polynomial degree is denoted by and the number of cells/grid points is denoted by . As a reference two black lines of slope 1 are drawn.
Mentions: In Fig. 1, we compare the error propagation for a Lagrange interpolation, a discontinuous Galerkin method, and the Fourier approximation using the initial value (4)u(0,x)=12+cosπx on the interval with periodic boundary conditions (the same interval and periodic boundary conditions are used for all simulations in this paper). As expected, initially the FFT method achieves a performance close to machine precision. Note, however, that the error growth is linear in the number of time steps. However, from a stochastic description of the round-off error one would expect an error growth proportional to the square root in the number of time steps. Let us postpone the detailed investigation of this issue until Section  5.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Innsbruck, Austria.

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.

No MeSH data available.