Improvements of Upstream, Semi-Lagrangian Numerical Advection Schemes

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  • 1 Institute of Meteorology and Geophysics, University of Vienna, Vienna, Austria
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Abstract

Semi-Lagrangian methods are common numerical techniques for the computation of the advection term in a nondivergent flow. They combine high accuracy and moderate computational requirements. Three aspects of these methods will be discussed. First, the condition of nondivergence must be carefully obeyed; examples of possible error sources are given. Second, the determination of the starting point of the backward trajectory can be improved considering a linear variation of the wind field around a grid point instead of the assumption of homogeneous velocity. It is shown both theoretically and by a suitable test that this improvement is necessary in a deformational flow in order to obtain satisfactory results. Examples are presented using cubic splines and a Fourier representation for interpolation of the advected quantity. Third, an improved method of making the scheme positive definite is introduced, which also removes the positive parts of 2Δx oscillations and avoids the potential instability of a similarly published method.

Abstract

Semi-Lagrangian methods are common numerical techniques for the computation of the advection term in a nondivergent flow. They combine high accuracy and moderate computational requirements. Three aspects of these methods will be discussed. First, the condition of nondivergence must be carefully obeyed; examples of possible error sources are given. Second, the determination of the starting point of the backward trajectory can be improved considering a linear variation of the wind field around a grid point instead of the assumption of homogeneous velocity. It is shown both theoretically and by a suitable test that this improvement is necessary in a deformational flow in order to obtain satisfactory results. Examples are presented using cubic splines and a Fourier representation for interpolation of the advected quantity. Third, an improved method of making the scheme positive definite is introduced, which also removes the positive parts of 2Δx oscillations and avoids the potential instability of a similarly published method.

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