Existence and stability of viscoelastic shock profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure

An a posterior error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type-norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a "mathematical" scheme derived from the weak formulation, and a phase-by-phase upstream weighting "engineering" scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient

An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow

International audienceIn this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type-norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a ''mathematical'' scheme derived from the weak formulation, and a phase-by-phase upstream weighting ''engineering'' scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient

Nonlinear dynamics of phase separation in thin films

We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. We introduce a repulsive substrate-film interaction potential and analyse the resulting fourth-order equations by constructing a Lyapunov functional, which, combined with the regularizing repulsive potential, gives rise to a positive lower bound for the free-surface height. The value of this lower bound depends on the parameters of the problem, a result which we compare with numerical simulations. While the theoretical lower bound is an obstacle to the rupture of a film that initially is everywhere of finite height, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips or `valleys' in free-surface height. We observe these valleys across zones where the concentration of the binary mixture changes sharply, indicating the formation of bubbles. Finally, we carry out numerical simulations without the repulsive interaction, and find that the film ruptures in finite time, while the gradient of the Cahn--Hilliard concentration develops a singularity.Comment: 26 pages, 20 figures, PDFLaTeX with RevTeX4 macros. A thorough analysis of the equations is presented in arXiv:0805.103