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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 1/2016

28.03.2015

High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations

verfasst von: Elena Celledoni, Bawfeh Kingsley Kometa, Olivier Verdier

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2016

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Abstract

We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge–Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139–164, 2009) for simpler convection–diffusion equations. We discuss the extension of these methods to the Navier–Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.
Vorheriger Artikel A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains
Nächster Artikel Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

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Fußnoten
1
We also assume that the characteristic paths are integrated to high accuracy.
 
2
Extension to higher order methods is straightforward (See [30] and references therein).
 
3
See 6.2 for the definitions of these norms.
 
4
The projection does not need to be orthogonal, but should be guaranteed not to compromise the order of the method, an orthogonal projection will have this property. The target of the orthogonal projection map on the discrete divergence-free subspace is the element of shortest distance to the point which is projected. Since \(y_{n+1}=\Pi Y_s\) and \(\Vert y(t_{n+1})-Y_s\Vert = {\mathcal {O}}(h^{r+1}),\) where \(r\) is the order of the integration method, then
$$\begin{aligned} \Vert y(t_{n+1})-y_{n+1}\Vert \le \Vert y(t_{n+1})-Y_s\Vert +\Vert Y_s-y_{n+1}\Vert = {\mathcal {O}}(h^{r+1}), \end{aligned}$$
because
$$\begin{aligned} \Vert y_{n+1}-Y_s\Vert \le \Vert y(t_{n+1})-Y_s\Vert . \end{aligned}$$
 
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Metadaten
Titel
High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations
verfasst von
Elena Celledoni
Bawfeh Kingsley Kometa
Olivier Verdier
Publikationsdatum
28.03.2015
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 1/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-015-0015-6

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