arXiv subjects

Pierre Benard

Publications and source records attributed to Pierre Benard.

3 recordsLinked to original sources

arXiv2003

Stability of leap-frog constant-coefficients semi-implicit schemes for the fully elastic system of Euler equations. Flat-terrain case

The aim of this paper is to investigate the response of this system/scheme in terms of stability in presence of explicitly treated residual terms, as it inevitably occurs in the reality of NWP. This sudy is restricted to the impact of thermal and baric residual terms (metric residual terms linked to the orography are not considered here). It is shown that conversely to what occurs with Hydrostatic Primitive Equations, the choice of the prognostic variables used to solve the system in time is of primary importance for the robustness with Euler Equations. For an optimal choice of prognostic variables, unconditionnally stable schemes can be obtained (with respect to the length of the time-step), but only for a smaller range of reference states than in the case of Hydrostatic Primitive Equations. This study also indicates that: (i) vertical coordinates based on geometrical height and on mass behave similarly in terms of stability for the problems examined here, and (ii) hybrid coordinates induce an intrinsic instability, the practical importance of which is however not completely elucidated in the theoretical context of this paper.

physics.ao-ph↗

arXiv2003

On the Use of a Wider Class of Linear Systems for the Design of Constant-Coefficients Semi-Implicit Time-Schemes in NWP

The linearization of the meteorological equations around a specified reference state, usually applied in NWP to define the linear system of constant-coefficients semi-implicit schemes, is outlined as an unnecessarily restrictive approach which may be detrimental in terms of stability. It is shown theoretically that an increased robustness can sometimes be obtained by choosing the reference linear system in a wider set of possibilities. The potential benefits of this new approach are illustrated in two simple examples. The advantage in robustness is not obtained at the price of an increased error or complexity.

physics.ao-ph↗

arXiv2003

Stability of Semi-Implicit and Iterative Centred-Implicit Time Discretizations for Various Equation Systems Used in NWP

The stability of classical semi-implicit scheme, and some more advanced iterative schemes recently proposed for Numerical Weather Prediction (NWP) purpose is examined. In all these schemes, the solution of the centred-implicit non-linear equation is approached by an iterative fixed-point algorithm, preconditioned by a simple, constant in time, linear operator. A general methodology for assessing analytically the stability of these schemes on canonical problems for a vertically unbounded atmosphere is presented. The proposed method is valid for all the equation systems usually employed in NWP. However, as in earlier studies, the method can be applied only in simplified meteorological contexts, thus overestimating the actual stability that would occur in more realistic meteorological contexts. The analysis is performed in the spatially-continuous framework, hence allowing to eliminate the spatial-discretisation or the boundary conditions as possible causes of the fundamental instabilities linked to the time-scheme itself. The general method is then shown concretely to apply to various time-discretisation schemes and equation-systems (namely shallow-water, and fully compressible Euler equations). Analytical results found in the literature are recovered from the proposed method, and some original results are presented.

physics.ao-ph↗