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On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

R. Herbin, W. Kheriji, J.-C. Latché (2014)

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

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution...

On stability of the P n mod / P n element for incompressible flow problems

Petr Knobloch (2006)

Applications of Mathematics

It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of n th order accuracy in the energy norm called P n elements. For n ≤ 3 we show that the stability condition holds if the velocity...

On the domain geometry dependence of the LBB condition

Evgenii V. Chizhonkov, Maxim A. Olshanskii (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The LBB condition is well-known to guarantee the stability of a finite element (FE) velocity - pressure pair in incompressible flow calculations. To ensure the condition to be satisfied a certain constant should be positive and mesh-independent. The paper studies the dependence of the LBB condition on the domain geometry. For model domains such as strips and rings the substantial dependence of this constant on geometry aspect ratios is observed. In domains with highly anisotropic substructures...

On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations

Vít Dolejší (2010)

Kybernetika

We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier–Stokes equations by the backward difference formula – discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss...

On the two-dimensional compressible isentropic Navier–Stokes equations

Catherine Giacomoni, Pierre Orenga (2002)

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

We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with γ = c p / c v = 2 . These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in...

On the two-dimensional compressible isentropic Navier–Stokes equations

Catherine Giacomoni, Pierre Orenga (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with γ = c p / c v = 2 . These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions...

Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids

Didier Bresch, Jonas Koko (2006)

International Journal of Applied Mathematics and Computer Science

We present a numerical simulation of two coupled Navier-Stokes flows, using ope-rator-split-ting and optimization-based non-overlapping domain decomposition methods. The model problem consists of two Navier-Stokes fluids coupled, through a common interface, by a nonlinear transmission condition. Numerical experiments are carried out with two coupled fluids; one with an initial linear profile and the other in rest. As expected, the transmission condition generates a recirculation within the fluid...

Optimal Convective Heat-Transport

Josef Dalík, Oto Přibyl (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The one-dimensional steady-state convection-diffusion problem for the unknown temperature y ( x ) of a medium entering the interval ( a , b ) with the temperature y min and flowing with a positive velocity v ( x ) is studied. The medium is being heated with an intensity corresponding to y max - y ( x ) for a constant y max > y min . We are looking for a velocity v ( x ) with a given average such that the outflow temperature y ( b ) is maximal and discuss the influence of the boundary condition at the point b on the “maximizing” function v ( x ) .

Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

Eberhard Bänsch, Klaus Deckelnick (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a finite element discretization by the Taylor–Hood element for the stationary Stokes and Navier–Stokes equations with slip boundary condition. The slip boundary condition is enforced pointwise for nodal values of the velocity in boundary nodes. We prove optimal error estimates in the H1 and L2 norms for the velocity and pressure respectively.

Optimal shape design in a fibre orientation model

Jan Stebel, Raino Mäkinen, Jukka I. Toivanen (2007)

Applications of Mathematics

We study a 2D model of the orientation distribution of fibres in a paper machine headbox. The goal is to control the orientation of fibres at the outlet by shape variations. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Existence of solutions both to the state and the optimization problem is analyzed and sensitivity analysis is performed. Further, discretization is done and a numerical example...

Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

A. Kadir Aziz, Donald A. French, Soren Jensen, R. Bruce Kellogg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations.

Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains

Cung The Anh, Dang Thanh Son (2015)

Annales Polonici Mathematici

We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality with a large class of non-autonomous external forces. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal...

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