Solution of inverse problems in contaminant transport with adsorption
Spatially-dependent and nonlinear fluid transport: coupling framework
We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation...
The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model
We construct a Roe-type numerical scheme for approximating the solutions of a drift-flux two-phase flow model. The model incorporates a set of highly complex closure laws, and the fluxes are generally not algebraic functions of the conserved variables. Hence, the classical approach of constructing a Roe solver by means of parameter vectors is unfeasible. Alternative approaches for analytically constructing the Roe solver are discussed, and a formulation of the Roe solver valid for general closure...
The generalized finite volume SUSHI scheme for the discretization of the peaceman model
We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later,...
The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section
We consider the Euler equations for compressible fluids in a nozzle whose cross-section is variable and may contain discontinuities. We view these equations as a hyperbolic system in nonconservative form and investigate weak solutions in the sense of Dal Maso, LeFloch and Murat [J. Math. Pures Appl.74 (1995) 483–548]. Observing that the entropy equality has a fully conservative form, we derive a minimum entropy principle satisfied by entropy solutions. We then establish the stability of a class...
The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems
We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem. We...
The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: II. The case of systems
We study the theoretical and numerical coupling of two hyperbolic systems of conservation laws at a fixed interface. As already proven in the scalar case, the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. We develop a detailed analysis of the coupling in the linear case. In the nonlinear case, we either use a linearized approach or a coupling method based on the solution of a Riemann problem....
Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson
When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al., SIAM J. Numer. Anal.43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p, less or equal to one, of a quantity that depends only on the mesh and on the advection ...
Transport of pollutant in shallow water : a two time steps kinetic method
The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other...
Transport of Pollutant in Shallow Water A Two Time Steps Kinetic Method
The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modeled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures – even in the case of a non flat bottom – the preservation of the steady state of a lake at rest, the non-negativity of the water height and the existence of an entropy inequality. On an other...
Vertex centred Discretization of Two-Phase Darcy flows on General Meshes
This paper concerns the discretization of multiphase Darcy flows, in the case of heterogeneous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems in [9], to multiphase Darcy flows....
Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.
Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system
We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.