A combined finite volume - finite element scheme for a low-Mach system involving a Joule term

Numerical Schemes for Semiconductors Energy-Transport Models

“…However, with such discretization, the rewriting of the scheme in dual entropy variables is not straightforward. Therefore, following [2], we propose an approximation of the Joule heating term which is based on its following reformulation:…”

A Finite Volume Method for a Convection-Diffusion Equation Involving a Joule Term

“…A first idea may be to consider a scheme also centered for this term, in order to increase the accuracy of the approximation, by following the piecewise discrete gradient introduced in [8], for instance. Nevertheless, we have shown in [2] that with this choice, a discrete maximum principle is only verified under very restrictive conditions on the initial data. A more effective approach is to consider an upwind discretization of the Joule term defined by:…”

“…Proof. The proof consists in an extension of the proof of Theorem 3.1 in [2] established in the case f ≡ 0, noticing in particular that ((u n+1 K − u n+1 K,σ ) + ) 2 = 0 if u n+1 K ≤ u n+1 K,σ . First, we prove that for any solution of (4), estimation (7) holds.…”

“…In this work, we present a finite volume scheme for the discretization of (1)-(2)-(3) which has been initially introduced in [2]. The aim of the present contribution is to give some results in the case f 0, and to investigate the efficiency of the derived scheme (as well as a variant one) on two original benchmarks.…”

“…The advantage of Scheme 1 is confirmed also considering smaller values of λ, or smaller magnitude of v or u ex . Finally, other schemes have been proposed and analyzed in [2]. Even if they are of order two in the case v = 0, they verify the maximum principle only under very restrictive conditions on the magnitude of M − m.…”