A conservative finite difference scheme for static diffusion equation.
A conservative spectral element method for the approximation of compressible fluid flow
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 controllability result for the -D isentropic Euler equation
A free boundary problem for compressible viscous fluids.
A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids
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
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
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
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
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 Lagrangian approach for the compressible Navier-Stokes equations
Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in , 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 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using...
A Multidimensional fluctuation splitting scheme for the three dimensional Euler equations
A multidimensional fluctuation splitting scheme for the three dimensional Euler equations
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...
A multigrid smoother for high Reynolds number flows.
A new domain decomposition method for the compressible Euler equations
In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains,...
A new numerical scheme for non uniform homogenized problems: application to the nonlinear Reynolds compressible equation.
A new technique in constructing closed-form solutions for nonlinear PDEs appearing in fluid mechanics and gas dynamics.
A note on critical times of quasilinear hyperbolic systems
In this paper the exact formula for the critical time of generating discontinuity (shock wave) in a solution of a 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 poroacoustic traveling waves under Darcy's law: Exact solutions
A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy's law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are...
A note on the free convection boundary layer on a vertical surface with prescribed heat flux at small Prandtl number.
A note on the relaxation-time limit of the isothermal Euler equations.