Maria Lukacova-Medvidova

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.

The paper is concerned with proving the equivalence of convexity or concavity properties of thermodynamic functions, such as energy and entropy, depending on different sets of variables. These variables are the basic thermodynamic state variables, specific state variables or the densities of state variables that are used in continuum mechanics. We prove results for transformations of variables and functions in conjunction with convexity properties. We are concerned with convexity, strict convexity, positive definite Hessian matrices and the analogous forms of concavity. The main results are equivalence relations for these properties between functions. These equivalences are independent of the equations of state since they only use general properties of them. The results can be used for instance to easily prove that the entropy density function for the Euler equations in conservative variables in three space dimensions is strictly concave or even has a negative definite Hessian matrix. Further, we show how various equations of state imply these properties and how these properties are relevant to mathematical analysis.

A conforming finite element scheme with mixed explicit–implicit time discretization for quasi-incompressible Navier–Stokes–Maxwell–Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier–Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell–Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties.

We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in probability. Convergence analysis combines results on the convergence of a deterministic FV method with stochastic compactness arguments due to Skorokhod and Gyöngy-Krylov.

We propose a new two-step selection criterion applicable to the dissipative measure--valued solutions of the Euler system of gas dynamics. The process consists of a successive maximisation of the entropy production rate and the total energy defect, i.e. maximisation of the turbulent energy. If the selected solution is a weak solution of the Euler system, then it is identified in the first step. Solutions selected in the second step are truly measure--valued maximising the energy defect. Accordingly, they are called turbulent solutions. The energy defect of turbulent solutions vanishes with growing time. The selected solutions depend in a Borel--measurable way on the initial data. In particular, they are almost continuously dependent on the initial data.

We model the flow behaviour of dense melts of flexible and semiflexible ring polymers in the presence of walls using a hybrid multiscale approach. Specifically, we perform molecular dynamics simulations and apply the Irving-Kirkwood formula to determine an averaged stress tensor for a macroscopic model. For the latter, we choose a Cahn-Hilliard-Navier-Stokes system with dynamic and no-slip boundary conditions. We present numerical simulations of the macroscopic flow that are based on a finite element method. In particular, we present detailed proofs of the solvability and the energy stability of our numerical scheme. Phase segregation under flow between flexible and semiflexible rings, as observed in the microscopic simulations, can be replicated in the macroscopic model by introducing effective attractive forces.

We propose new fully discrete third-order accurate Active Flux and WENO methods based on truly multidimensional evolution operators for the two-dimensional acoustic equations. Building on the method of bicharacteristics, several approximate evolution operators are derived that yield an improved stability of the resulting schemes. A linear stability analysis is applied to determine the maximal CFL number. The schemes are tested extensively on both continuous and discontinuous problems, confirming their robustness and accurate approximation even on coarse grids.