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We consider consistent, conservative-form, monotone nite di erence schemes for nonlinear convection-di usion equations in one space dimension. Since we allow the di usion term to be strongly degenerate, solutions can be discontinuous and are in general not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin 30] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase ow equation and hyperbolic conservation laws. The di erence schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone di erence approximations of conservation laws, the main di culty in obtaining a similar convergence theory in the present context is to show that the approximations are L 1 Lipschitz continuous in the time variable (this is trivial for conservation laws). This continuity result is in turn intimately related to the regularity properties possessed by the (strongly degenerate) discrete di usion term. We provide the necessary regularity estimates on the di usion term by deriving and carefully analysing a linear di erence equation satis ed by the numerical ux of the di erence schemes. x1. Introduction. We are interested in monotone nite di erence approximations of nonlinear, possibly strongly degenerate, convection-di usion problems of form (1) (@ t u + @ x f(u) = @ x (k(u)@ x u); (x; t) 2 Q T R h0; Ti ; k(u) 0; u(x; 0) = u 0 (x); where the initial condition u 0 (x), the convection ux f(u) and the di usion ux k(u) 0 are given, su ciently regular functions. Convection-di usion equations arise in a variety of applications, among others turbulence, tra c ow, nancial modelling, front propagation, two phase ow in oil reservoirs, and in models describing certain sedimentation processes. When (1) is non-degenerate, i.e., k(u) > 0, it is well known that (1) admits a unique classical solution 21]. This contrasts with the degenerate case where k(u) may vanish for some values of u. A simple example of a degenerate equation is the porous medium equation, (2) @ t u = @ x (u m) ; m > 1; 1991 Mathematics Subject Classi cation. 65M12, 35K65, 35L65. Key words and phrases. Degenerate convection-di usion equations, BV solutions, entropy condition, monotone nite di erence schemes, convergence. Karlsen has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).

We present a multi-stage centred scheme, of the kind proposed by Toro [Appl. Numer. Math. 56 (2006) 1464], for numerically resolving the simultaneous flow of two fluids through a transport pipeline. This model contains non-conservative terms in both the temporal and spatial derivatives, and an extension of the standard numerical framework for conservation laws is needed. In this paper, we rewrite the model in an equivalent mathematical form, eliminating the nonconservative time-derivatives. This allows us to use the framework described by Parés [SIAM J. Numer. Anal. 44 (2006) 300]. We develop FORCE and MUSTA-type schemes which are consistent with Parés' formalism. Numerical simulations demonstrate a high degree of stability of our proposed schemes. Comparisons with the Roe and Rusanov schemes indicate that convergence to near-identical solutions are obtained when the non-conservative terms are discretized with respect to the same evaluation of the path-dependent integrals. However, if the schemes are not mutually formally path-consistent in the sense of Parés, different converged solutions are obtained.

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We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.

The purpose of this paper is to explore a viscous two-phase liquid-gas model relevant for well and pipe flow. Our approach relies on applying suitable modifications of techniques previously used for studying the single-phase isothermal Navier-Stokes equations. A main issue is the introduction of a novel two-phase variant of the potential energy function needed for obtaining fundamental a priori estimates. We derive an existence result for weak solutions in a setting where transition to single-phase flow is guaranteed not to occur when the initial state is a true mixture of both phases. Some numerical examples are also included in order to demonstrate characteristic behavior of solutions. In particular, we illustrate how two-phase flow is genuinely different compared to single-phase flow concerning the behavior of an initial mass discontinuity.

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