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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 solution of (11) is computed for the final time . The analytical solution is shown on the top left. The numerical solution is computed using a splitting approach. The results for the FFT based method (top right), the discontinuous Galerkin (dG) method (bottom left), and the Lagrange and spline interpolations (bottom right) are shown. The black lines have slope 1 and are shown for comparison.
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f000040: The solution of (11) is computed for the final time . The analytical solution is shown on the top left. The numerical solution is computed using a splitting approach. The results for the FFT based method (top right), the discontinuous Galerkin (dG) method (bottom left), and the Lagrange and spline interpolations (bottom right) are shown. The black lines have slope 1 and are shown for comparison.

Mentions: In this section we consider the advection equation supplemented by a (position dependent) source term. That is, we consider(11)∂tu(t,x)+∂xu(t,x)=s(x), where the source term is chosen as and periodic boundary conditions are imposed. The solution can be easily determined by the method of characteristics u(t,x)=u0(x−t)+∫0ts(x−t+σ)dσ, which for the initial value given in (4) is plotted in Fig. 8 (top left).


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

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

The solution of (11) is computed for the final time . The analytical solution is shown on the top left. The numerical solution is computed using a splitting approach. The results for the FFT based method (top right), the discontinuous Galerkin (dG) method (bottom left), and the Lagrange and spline interpolations (bottom right) are shown. The black lines have slope 1 and are shown for comparison.
© Copyright Policy - CC BY
Related In: Results  -  Collection

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

f000040: The solution of (11) is computed for the final time . The analytical solution is shown on the top left. The numerical solution is computed using a splitting approach. The results for the FFT based method (top right), the discontinuous Galerkin (dG) method (bottom left), and the Lagrange and spline interpolations (bottom right) are shown. The black lines have slope 1 and are shown for comparison.
Mentions: In this section we consider the advection equation supplemented by a (position dependent) source term. That is, we consider(11)∂tu(t,x)+∂xu(t,x)=s(x), where the source term is chosen as and periodic boundary conditions are imposed. The solution can be easily determined by the method of characteristics u(t,x)=u0(x−t)+∫0ts(x−t+σ)dσ, which for the initial value given in (4) is plotted in Fig. 8 (top left).

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.