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Well-posed lateral boundary conditions for spectral semi-implicit semi-Lagrangian schemes : tests in a one-dimensional model

(2009) MONTHLY WEATHER REVIEW. 137(1). p.315-330
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Abstract
The aim of this paper is to investigate the feasibility of well-posed lateral boundary conditions in a Fourier spectral semi-implicit semi-Lagrangian one-dimensional model. Two aspects are analyzed: (i) the complication of designing well-posed boundary conditions for a spectral semi-implicit scheme and (ii) the implications of such a lateral boundary treatment for the semi-Lagrangian trajectory computations at the lateral boundaries. Straightforwardly imposing boundary conditions in the gridpoint-explicit part of the semi-implicit time-marching scheme leads to numerical instabilities for time steps that are relevant in today's numerical weather prediction applications. It is shown that an iterative scheme is capable of curing these instabilities. This new iterative boundary treatment has been tested in the framework of the one-dimensional shallow-water equations leading to a significant improvement in terms of stability. As far as the semi-Lagrangian part of the time scheme is concerned, the use of a trajectory truncation scheme has been found to be stable in experimental tests, even for large values of the advective Courant number. It is also demonstrated that a well-posed buffer zone can be successfully applied in this spectral context. A promising (but not easily implemented) alternative to these three above-referenced schemes has been tested and is also presented here.
Keywords
LIMITED-AREA MODEL, SHALLOW-WATER EQUATIONS, BAROCLINIC WAVES, FORMULATION, PREDICTION, SYSTEMS, UPDATE, NWP

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MLA
Voitus, Fabrice, Piet Termonia, and Pierre Benard. “Well-posed Lateral Boundary Conditions for Spectral Semi-implicit semi-Lagrangian Schemes : Tests in a One-dimensional Model.” MONTHLY WEATHER REVIEW 137.1 (2009): 315–330. Print.
APA
Voitus, F., Termonia, P., & Benard, P. (2009). Well-posed lateral boundary conditions for spectral semi-implicit semi-Lagrangian schemes : tests in a one-dimensional model. MONTHLY WEATHER REVIEW, 137(1), 315–330.
Chicago author-date
Voitus, Fabrice, Piet Termonia, and Pierre Benard. 2009. “Well-posed Lateral Boundary Conditions for Spectral Semi-implicit semi-Lagrangian Schemes : Tests in a One-dimensional Model.” Monthly Weather Review 137 (1): 315–330.
Chicago author-date (all authors)
Voitus, Fabrice, Piet Termonia, and Pierre Benard. 2009. “Well-posed Lateral Boundary Conditions for Spectral Semi-implicit semi-Lagrangian Schemes : Tests in a One-dimensional Model.” Monthly Weather Review 137 (1): 315–330.
Vancouver
1.
Voitus F, Termonia P, Benard P. Well-posed lateral boundary conditions for spectral semi-implicit semi-Lagrangian schemes : tests in a one-dimensional model. MONTHLY WEATHER REVIEW. 2009;137(1):315–30.
IEEE
[1]
F. Voitus, P. Termonia, and P. Benard, “Well-posed lateral boundary conditions for spectral semi-implicit semi-Lagrangian schemes : tests in a one-dimensional model,” MONTHLY WEATHER REVIEW, vol. 137, no. 1, pp. 315–330, 2009.
@article{6855717,
  abstract     = {The aim of this paper is to investigate the feasibility of well-posed lateral boundary conditions in a Fourier spectral semi-implicit semi-Lagrangian one-dimensional model. Two aspects are analyzed: (i) the complication of designing well-posed boundary conditions for a spectral semi-implicit scheme and (ii) the implications of such a lateral boundary treatment for the semi-Lagrangian trajectory computations at the lateral boundaries. 
Straightforwardly imposing boundary conditions in the gridpoint-explicit part of the semi-implicit time-marching scheme leads to numerical instabilities for time steps that are relevant in today's numerical weather prediction applications. It is shown that an iterative scheme is capable of curing these instabilities. This new iterative boundary treatment has been tested in the framework of the one-dimensional shallow-water equations leading to a significant improvement in terms of stability. 
As far as the semi-Lagrangian part of the time scheme is concerned, the use of a trajectory truncation scheme has been found to be stable in experimental tests, even for large values of the advective Courant number. It is also demonstrated that a well-posed buffer zone can be successfully applied in this spectral context. A promising (but not easily implemented) alternative to these three above-referenced schemes has been tested and is also presented here.},
  author       = {Voitus, Fabrice and Termonia, Piet and Benard, Pierre},
  issn         = {0027-0644},
  journal      = {MONTHLY WEATHER REVIEW},
  keywords     = {LIMITED-AREA MODEL,SHALLOW-WATER EQUATIONS,BAROCLINIC WAVES,FORMULATION,PREDICTION,SYSTEMS,UPDATE,NWP},
  language     = {eng},
  number       = {1},
  pages        = {315--330},
  title        = {Well-posed lateral boundary conditions for spectral semi-implicit semi-Lagrangian schemes : tests in a one-dimensional model},
  url          = {http://dx.doi.org/10.1175/2008MWR2539.1},
  volume       = {137},
  year         = {2009},
}

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