Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

The Brazilian Zero Hunger social program implications in achieving sustainable development

This article aims to analyse the progress taken by Brazil towards the accomplishment of a sustainable development, mainly highlighting the success of the Zero Hunger social programme in achieving the elimination of starvation. As one of the essential pillars to the sustainable development, the State’s social performance demands actions directed to the elimination of extreme poverty and hunger, establishing a basis for equitable growth. It is necessary to clarify the juridical and constitutional framework of Brazil, aiming to emphasise the Zero Hunger importance in achieving the Brazilian Republic’s goals and reaching an international pattern on sustainable development. Also, it will be stressed that the international order influences Brazilian domestic law and is one of the main aspects for the Zero Hunger program development.