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A conservative spectral element method for the approximation of compressible fluid flow

Kybernetika

A method to approximate the Euler equations is presented. The method is a multi-domain approximation, and a variational form of the Euler equations is found by making use of the divergence theorem. The method is similar to that of the Discontinuous-Galerkin method of Cockburn and Shu, but the implementation is constructed through a spectral, multi-domain approach. The method is introduced and is shown to be a conservative scheme. A numerical example is given for the expanding flow around a point...

A Lagrangian approach for the compressible Navier-Stokes equations

Annales de l’institut Fourier

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in ${ℝ}^{n}$, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from $0$ and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using...

A note on critical times of $2×2$ quasilinear hyperbolic systems

Aplikace matematiky

In this paper the exact formula for the critical time of generating discontinuity (shock wave) in a solution of a $2×2$ quasilinear hyperbolic system is derived. The applicability of the formula in the engineering praxis is shown on one-dimensional equations of isentropic non-viscous compressible fluid flow.

A note on the relaxation-time limit of the isothermal Euler equations.

Boundary Value Problems [electronic only]

A numerical method for unsteady flows

Applications of Mathematics

A high resolution finite volume method for the computation of unsteady solutions of the Euler equations in two space dimensions is presented and validated. The scheme is of Godunov-type. The first order part of the flux function uses the approximate Riemann problem solver of Pandolfi and here a new derivation of this solver is presented. This construction paves the way to understand the conditions under which the scheme satisfies an entropy condition. The extension to higher order is done by applying...

A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations

Annales de l'I.H.P. Analyse non linéaire

A remark on the smoothness of bounded regions filled with a steady compressible and isentropic fluid

Applications of Mathematics

For convenient adiabatic constants, existence of weak solutions to the steady compressible Navier-Stokes equations in isentropic regime in smooth bounded domains is well known. Here we present a way how to prove the same result when the bounded domains considered are Lipschitz.

A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum

Applications of Mathematics

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_{i}$ of the fluids and their velocity fields ${u}^{\left(i\right)}$ are prescribed at infinity: ${\rho }_{i}{|}_{\infty }={\rho }_{i\infty }>0$, ${u}^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_{i}\equiv {\rho }_{i\infty }$, ${u}^{\left(i\right)}\equiv 0$, $i=1,2$.

A Van Leer finite volume scheme for the Euler equations on unstructured meshes

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

About steady transport equation I – ${L}^{p}$-approach in domains with smooth boundaries

Commentationes Mathematicae Universitatis Carolinae

We investigate the steady transport equation $\lambda z+w·\nabla z+az=f,\phantom{\rule{1.0em}{0ex}}\lambda >0$ in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions $w,\phantom{\rule{0.166667em}{0ex}}a$ are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular stress is put onto the problem to extend the results to as less regular vector fields $w,\phantom{\rule{0.166667em}{0ex}}a$, as possible (conserving the requirement of...

About steady transport equation. II: Schauder estimates in domains with smooth boundaries.

Portugaliae Mathematica

About the Resolvent of an Operator from Fluid Dynamics.

Mathematische Zeitschrift

Abrupt and smooth separation of free boundaries in flow problems

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Acoustic-gravity waves in a viscous and thermally conducting isothermal atmosphere. III: For arbitrary Prandtl number.

International Journal of Mathematics and Mathematical Sciences

Acoustic-gravity waves in a viscous and thermally conducting isothermal atmosphere. I: For large Prandtl number.

International Journal of Mathematics and Mathematical Sciences

Acoustic-gravity waves in a viscous and thermally conducting isothermal atmosphere. II: For small Prandtl number.

International Journal of Mathematics and Mathematical Sciences

An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit

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

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on $x$. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low...

An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit

ESAIM: Mathematical Modelling and Numerical Analysis

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss...

An existence proof for the stationary compressible Stokes problem

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation of the continuity equation.

An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

ESAIM: Mathematical Modelling and Numerical Analysis

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of...

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