A MUSCL method satisfying all the numerical entropy inequalities

Abstract

Abstract. We consider here second-order finite volume methods for onedimensional scalar conservation laws. We give a method to determine a slope reconstruction satisfying all the exact numerical entropy inequalities. It avoids inhomogeneous slope limitations and, at least, gives a convergence rate of ∆x 1/2 . It is obtained by a theory of second-order entropic projections involving values at the nodes of the grid and a variant of error estimates, which also gives new results for the first-order Engquist-Osher scheme.

Kruzkov’s estimates for scalar conservation laws revisited

“…This strategy was initiated in a paper by the authors and C. Bourdarias [2]. This approach is shown to be successful in the most classical situations, which we develop independently in § §3, 4, 5.…”

“…For numerical schemes, the assumption (2.2) on α G is somehow related to an estimate on the time modulus of continuity of u. In fact, α G is comparable with ∆t ∂ t u, with ∆t the time step (see [2]). However, when no such estimate is available, it is possible to get an estimate of u − v L 1 t,x with only the regularity of α G stated in (1.4) and an estimate of α G only for small t. In order to obtain this result, just average (2.4) with respect to T and then observe that when averaged, the last inequality of (2.28) in the proof below can be replaced by a suitable one since |χ (t)| ≤ Y ε (t)+Y ε (t−T ).…”

Entropic approximation in kinetic theory

“…Then, we follow the idea of Levermore that consists in reconstructing a function G from the moments ρ through an entropy minimization principle [11]. Notice that such a principle is also used within the context of hyperbolic systems, that is, at a macroscopic level to define entropic projections suited to numerical approximation [3,4]. However in that context the space on which the projection is defined (piecewise linear functions) is set a priori while here it is not and is rather defined a posteriori.…”

Stability of reconstruction schemes for scalar hyperbolic conservations laws

“…-cell function for every constant-in-cell function u, the resulting scheme can be a MUSCL scheme as studied in [3]. Nevertheless, we do not use such characterizations of the reconstruction form in the following.…”

“…Among the wide amount of studies, the reader can refer to the classical references [27,28], to [13] for a general study of discrete entropy conditions, to [9] for the geometric limiters theory (slope limiters), to [25] for the flux limiter theory, and to [5] for its extension to the Euler system [4,20] for the study of MUSCL schemes and entropy. One can read [3] for a precise study of links between geometric reconstruction and decrease of numerical entropy, and [18] for high order approximation with entropy inequalities. We also also mention [22] for a general study of convergence and order.…”