We consider two-phase flow problems in porous media with overshooting waves. If higher order effects are neglected, such problems are governed by a hyperbolic conservation law with non-convex flux function as macroscale model. It is well known, that there can be multiple weak solutions. To ensure uniqueness, we use on the microscale either a kinetic relation or in a second approach an extended system, taking rate-dependent capillary pressure effects into account. We present a new multidimensional mass-conserving numerical method to solve the macroscale model. The method belongs to the class of Heterogeneous Multiscale Methods in the sense of E&Engquist (2003) and is L ∞-stable. A key part of the approximation is a novel numerical flux function for the multidimensional setting, which captures undercompressive waves and generalizes the one-dimensional approach of Boutin et al. (2008). Furthermore, we improve the overall computational complexity, using a data-based approach. Finally, we validate the numerical method and test it on several infiltration problems.

We consider weak solutions of hyperbolic conservation laws as singular limits of solutions for associated complex regularized problems. We are interested in situations such that undercompressive (Non-Laxian) shock waves occur in the limit. In this setting one can view the conservation law as a macroscale formulation while the regularization can be understood as the microscale model.With this point of view it appears natural to solve the macroscale model by a heterogeneous multiscale approach in the sense of E&Engquist [7]. We introduce a new mass-conserving numerical method based on this concept and test it on scalar model problems. This includes applications from phase transition theory as well as from two-phase flow in porous media.2000 Mathematics Subject Classification. Primary: 35L65.

We consider the infi ltra on of a we ng fl uid into a homogeneous porous medium and the forma on of preferen al fl ow paths with satura on overshoots. These are caused by dynamic capillary pressure eff ects which can be modeled by the rate-dependent approach of Hassanizadeh and Gray. To track the overshoot wave numerically, we propose an extended Heterogeneous Mul scale Method. In the approach, the ratedependent model takes the role of a microscale model. The algorithm is applied to several infi ltra on problems.Abbrevia ons: HMM, heterogeneous mul scale method.

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We consider a non-local regularization of nonlinear hyperbolic conservation laws in several space variables. The regularization is motivated by the theory of phase dynamics and is based on a convolution operator. We formulate the initial value problem and begin by deriving a priori estimates which are independent of the regularization parameter. Following Hwang and Tzavaras we establish a kinetic decomposition associated with the problem under consideration, and we conclude that the sequence of solutions generated by the non-local model converges to a weak solution of the corresponding hyperbolic problem. Depending on the scaling introduced in the non-local dispersive term, this weak limit is either a classical Kruzkov solution satisfying all entropy inequalities or, more interestingly, a nonclassical entropy solution in the sense defined by LeFloch, that is, a weak solution satisfying a single entropy inequality and containing undercompressive shock waves possibly selected by a kinetic relation. Finally, we illustrate our analytical conclusions with numerical experiments in one spatial variable.