ERROR ESTIMATES FOR THE STAGGERED LAX–FRIEDRICHS SCHEME ON UNSTRUCTURED GRIDS ∗

Abstract. Staggered grid finite volume methods (also called central schemes) were introduced in one dimension by Nessyahu and Tadmor in 1990 in order to avoid the necessity of having information on solutions of Riemann problems for the evaluation of numerical fluxes. We consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption. We interpret the staggered Lax–Friedrichs scheme as a three-step method consisting of a prolongation step onto a finer intersection grid, a finite volume step with an arbitrarily good numerical flux (e.g., Godunov flux) on the intersection grid, followed by an averaging step such that the calculation of numerical fluxes reduces to evaluations of the continuous flux. Using this point of view, we prove an a posteriori error estimate and an a priori error estimate in the L 1-norm in space and time which is of order h 1/4, where h is a mesh-size parameter. Hence, we recover for the staggered Lax–Friedrichs scheme the same order of convergence as for upwind finite volume methods on a fixed grid. AMS subject classifications. 65M15, 75M12, 35L6

Manger la chair de son ennemi

International audienceThe goal of this article is to study, through the corpus of Homeric formulas, what might be called a “Homeric Encyclopedia” of food. Most of the formulas provide us with a coherent image of mankind eating bread and cooked animal meat, as opposed to animals eating human flesh and to gods eating nectar and ambrosia. However, some troubling passages show that Achilles, face-to-face with Hector, wishes that he could eat him. Moreover, queen Hecuba in book 24 wishes that she could eat Achilles’ liver. A single passage in the first half of the Iliad shows a similar vow, perhaps ironically attributed by Zeus to Hera, with the rare form βεβρώθοις. We link these passages to two others, where a weapon, namely a spear, is said to be eager for (human) flesh. Though not as common in the texts as the more usual formulas, these forms take part in the formulaic corpus, and evoke a more savage image of the Homeric world, where the desire to eat one’s enemy is not masked by the usual linguistic conventions.À partir des formules homériques se dessine une “Encyclopédie homérique” de la nourriture. La plupart des formules montrent une image cohérente des humains mangeant du pain et de la viande cuite, en opposition aux animaux nourris de chair humaine et aux dieux nourris de nectar et d’ambroisie. Pourtant, certains passages troublants montrent qu’Achille face à Hector fait le vœu de le manger cru. Hécube dans le chant 24 souhaite de manger cru le foie d’Achille. Dans la première partie de l’Iliade, un seul passage montre un vœu comparable, peut-être ironiquement attribué à Héra par Zeus, avec la forme rare βεβρώθοις. Nous relions ces passages à d’autres dans lesquels une arme, une lance, est dite être avide de chair humaine. Même si elles ne sont pas aussi fréquentes dans le texte que celles du premier groupe, ces formes font partie du corpus formulaire, et entraînent une image plus sauvage du monde homérique, dans laquelle le désir de manger son ennemi n’est pas masqué sous les conventions usuelles du langage