Narrow Results

Stencil codes such as the Jacobi, Gauß-Seidel, and red-black Gauß-Seidel kernels are among the most time-consuming routines in many scientific and engineering applications. The performance of these codes critically depends on an efficient usage of caches, and can be improved by tiling. Several tiling schemes have been suggested in the literature; this paper gives an overview and comparison. Then, in the main part, we prove a lower bound on the number of cold and capacity misses. Finally, we analyze a particular tiling scheme, and show that it is off the lower bound by a factor of at most ten. Our results show up limitations to the speedup that can be gained by future research.

We derive a new relaxation method for the compressible Navier–Stokes equations endowed with general pressure and temperature laws compatible with the existence of an entropy functional and Gibbs relations. Our method is an extension of the energy relaxation method introduced by Coquel and Perthame for the Euler equations. We first introduce a consistent splitting of the diffusion fluxes as well as a global temperature for the relaxation system. We then prove that under the same subcharacteristic conditions as for the relaxed Euler equations and for a specific form of the global temperature and the heat flux splitting, the stability of the relaxation system may be obtained from the non-negativity of a suitable entropy production. A first-order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method allowing for a straightforward use of Navier–Stokes solvers designed for ideal gases as well as numerical results illustrating the accuracy of the proposed algorithm.

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.

The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl. Math.48, 235, (1995)] in order to approximate the conservation law ∂_tu + ∂_xf(u) = 0 for any flux function f ∈ 𝒞¹ (ℝ;ℝ). In this paper, we propose an alternative relaxation technique for scalar conservation laws of the form ∂_tu + ∂_xu(1 - u)g(u) = 0, where g ∈ 𝒞¹ ([0, 1]; ℝ) and 0 ∉ g(]0, 1[). We extend this new philosophy to an arbitrary flux function f whenever possible. Unlike the semilinear approach, the new relaxation strategy does not involve any tuning parameter, but makes use of the Born–Infeld system. Another advantage of this method is that it enables us to achieve a maximum principle on the velocities w = (1 - u)g and z = -ug, which turns out to be a physically interesting and helpful feature in the context of some two-phase flow problems.