A high-order finite volume method for systems of conservation laws—Multi-dimensional Optimal Order Detection (MOOD)

In this paper, we investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Multi-dimensional Optimal Order Detection (MOOD) breaks away from classical limitations employed in high-order methods. The proposed method consists of detecting problematic situations after each time update of the solution and of reducing the local polynomial degree before recomputing the solution. As multi-dimensional MUSCL methods, the concept is simple and independent of mesh structure. Moreover MOOD is able to take physical constraints such as density and pressure positivity into account through an “a posteriori” detection. Numerical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of this approach.     

The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems

The Multidimensional Optimal Order Detection (MOOD) method for two‐dimensional geometries has been introduced by the authors in two recent papers. We present here the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids, and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high order of accuracy is reached on smooth solutions, whereas spurious oscillations near singularities are prevented. At last, the intrinsic positivity‐preserving property of the MOOD method is confirmed in 3D, and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared with existing high‐order Finite Volume methods.Copyright © 2013 John Wiley & Sons, Ltd.     

L ∞ stability of the MUSCL methods

We present a general L ^∞ stability result for generic finite volume methods coupled with a large class of reconstruction for hyperbolic scalar equations. We show that the stability is obtained if the reconstruction respects two fundamental properties: the convexity property and the sign inversion property. We also introduce a new MUSCL technique named the multislope MUSCL technique based on the approximations of the directional derivatives in contrast to the classical piecewise reconstruction, the so-called monoslope MUSCL technique, based on the gradient reconstruction. We show that under specific constraints we shall detail, the two MUSCL reconstructions satisfy the convexity and sign inversion properties and we prove the L ^∞ stability.     

A Two-dimensional Stationary Induction Heating Problem

We consider in this paper a system of equations modelling a steady-state induction heating process for ‘two-dimensional geometries’. Existence of a solution is stated in W^1,p(Ω) Sobolev spaces and is derived using the Leray–Schauder's fixed point theory. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Capillarity-dissolution system for a two-dimensional geometry

We consider a system composed of two fluids in contact with a solid where one of these fluids dissolves the solid material. Both the dissolution process and the capillary phenomena play a role in the system evolution, which is analyzed on the basis of stability arguments for a two-dimensional geometry.     

Formation of adhesive contacts: Spreading versus dewetting

A soft bead (radius R _b) is pressed with a force F against a hydrophobic glass plate through a water drop (“wet” JKR set-up). We observe with a fast camera the growth of the contact zone bridging the rubber bead to the glass. Depending on the approach velocity V, two regimes are observed : i) at large V a liquid film is squeezed at the interface and dewets by nucleation and growth of a dry contact; ii) at low velocities, the bead remains nearly spherical. As it comes into contact, the rubber bead spreads on the glass with a characteristic time (in the range of one millisecond) τ ≈ ηR _b ²/F, where η is the liquid viscosity. The laws of spreading are interpreted by a balance of global mechanical and viscous forces. Received: 22 December 2002 / Accepted: 24 March 2003 / Published online: 29 April 2003 RID="a" ID="a"e-mail: brochard@curie.fr     

Numerical Simulation of Darcy and Forchheimer Force Distribution in a HBC Fuse

We present the breaking of a short-circuit current in a HBC fuse simulation based on an isentropic non-stationary model in a porous medium for a one dimensional geometry. The fluid flow is affected by the nature of the gas and by the morphology of the silica sand. To model the gas–silica sand interaction, we introduce two classical laws: the Darcy's law due to the viscous interaction and the Forchheimer's law due to the inertial force. Numerical simulations with realistic physical parameters have been performed using a finite volume scheme with a fractional step technique. We show the evolution of Darcy and Forchheimer forces during time and according to the position in the fuse. We place in prominent position the fact that either force is predominant in the fuse according to the time and the position which justifies a numerical treatment to cover all the situations.     

Nucleation of wave defects and formation of optical localized structures in bi-directional ring lasers

This work is a contribution to the calculation of transport coefficients for nitrogen, hydrogen, Mars and Titan atmospheres plasmas. The calculation which assumes local thermodynamic equilibrium is performed using Lennard-Jones potential and simple combining rules for the potential parameters for neutral-neutral interaction. A discussion is made for the other interactions: neutral-charged, charged-charged and electron-neutral. The results are compared with those of published theoretical studies for a temperature up to 30 000 K.     

<Emphasis Type="Italic">a posteriori</Emphasis> stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations.