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A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme

Hyam Abboud, Toni Sayah (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH...

A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

T. Haga, H. Gao, Z. J. Wang (2011)

Mathematical Modelling of Natural Phenomena

The newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form....

A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport

Manuel Jesús Castro Díaz, Enrique Domingo Fernández-Nieto, Tomás Morales de Luna, Gladys Narbona-Reina, Carlos Parés (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different HLLC solvers. Some results concerning the non-negativity...

A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport

Manuel Jesús Castro Díaz, Enrique Domingo Fernández-Nieto, Tomás Morales de Luna, Gladys Narbona-Reina, Carlos Parés (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-type approximate Riemann solver for a hyperbolic nonconservative PDE system arising in a turbidity current model. The main difficulties come from the nonconservative nature of the system. A general strategy to derive simple approximate Riemann solvers for nonconservative systems is introduced, which is applied to the turbidity current model to obtain two different...

A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Thierry Gallouët, Jean-Marc Hérard, Nicolas Seguin (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional...

A hybrid scheme to compute contact discontinuities in one-dimensional Euler systems

Thierry Gallouët, Jean-Marc Hérard, Nicolas Seguin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density...

A mixed formulation of a sharp interface model of stokes flow with moving contact lines

Shawn W. Walker (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking...

A modified Cayley transform for the discretized Navier-Stokes equations

K. A. Cliffe, T. J. Garratt, Alastair Spence (1993)

Applications of Mathematics

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...

A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

Jérôme Bastin, Gilbert Rogé (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The fluctuation splitting schemes were introduced by Roe in the beginning of the 80's and have been then developed since then, essentially thanks to Deconinck. In this paper, the fluctuation splitting schemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of the three dimensional Euler equations is presented. This decomposition leads to an acoustic subsystem and two scalar advection equations, one of them being the entropy advection. Thanks to this decomposition, the two scalar...

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