Abstract. We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes.

We consider scalar hyperbolic conservation laws in spatial dimension d ≥ 1 d\geq 1 with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and give sufficient conditions on the initial data that ensure the existence of statistical moments of any order k k of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of the ensemble average of the random entropy solutions as well as of their k k -point space-time correlation functions. These schemes are shown to obey the same accuracy vs. work estimate as a single application of the finite volume solver for the corresponding deterministic problem. Numerical experiments demonstrating the efficiency of these schemes are presented. In certain cases, statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

There is no theory for the initial value problem for compressible flows in two space dimensions once shocks show up, much less in three space dimensions. This is a scientific scandal and a challenge." P. D. Lax, 2007 Gibbs Lecture [48] Abstract Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in [17,18] have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes may not necessarily converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions.Furthermore, we propose a more general notion, that of entropy measure valued solutions, as an appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.