Research Theme 1: Measure Valued Solutions and Uncertainty Quantification
The problem of weak-strong uniqueness of measure-valued solutions to a classical system of conservation laws arising in elastodynamics and polyconvex thermoelasticity was resolved (UNIVAQ, UOS, FORTH and KAUST). Issues related to ergodicity of spherically symmetric fluid flows related to black holes are addressed (SU) New computational methods were proposed and studied. First, a new paradigm for UQ in tsunami simulations has been proposed by the UMA and ETH team. Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs (SU and ETH) were designed and analysed. In connection with uncertainty quantification for conservation laws, the UIO node has proved results regarding the convergence rate for upwind schemes applied to conservation laws with discontinuous coefficients. Recent results by our groups in UIO and ETH suggest the use of a stronger notion of measure valued solutions, namely of statistical solutions, where one considers solutions which are probability measures on function spaces. Also, corresponding numerical algorithms for the computation of statistical solutions were devised based on Monte Carlo sampling. It is crucial to guarantee that the deterministic solvers retain certain stability and entropy consistency properties. Such solvers which in addition are compatible with the notion of entropic measure valued solutions were proposed and analysed by the UOS and FORTH teams. Also, an alternative approach to computation of measure valued solutions of HCL was proposed by the UOS team based on the introduction of discrete and generalised kinetic models.
Research Theme 2: Propagating Interfaces
Treatment of interfaces is one of the main topics of the project. On the numerical side, the UNICT team developed level set-ghost point methods on Cartesian meshes for the treatment of elliptic problem with jumps across interfaces and for the numerical solution of the Euler equation of gas dynamics. The UNIVAQ team addressed issues related to the existence and regularity of solutions and scale limit analysis. The SU team studied several analytic issues related to propagating interfaces. On the computational side, new schemes were proposed and analysed for hyperbolic systems in nonconservative form (ETH, UMA, SU), computational methods of non-conservative products using Roe-type path conservative scheme (SU). Also, a new numerical strategy for relativistic fluid was developed (SU, UMA). The UIO node has also studied numerical methods for scalar conservation laws on graphs, with a view towards modelling traffic on a network of roads. Further, the application of a spectral method to the Euler equations was studied (ETH, UIO). The UIO team has worked as well on proving convergence of entropy stable methods of multi-dimensional conservation laws and for so-called non-local conservation laws.
Research Theme 3: Models and Methods across scales
The UNICT team had significant progress on the development of semi-implicit numerical methods for hyperbolic systems. UNIVAQ team was focused on the scale analysis for fluid dynamical models used in different context. RWTH team is working on the development of a novel Lagrangian method for the simulation of compressible fluid flows. Also, RWTH team has proposed a new approach for transport problems with non-Lipschitz coefficients. A very promising approach for data assimilation and reduced modeling was initiated (RWTH, SU). Entropy diminishing schemes were introduced for conservation laws (UOS) and for degenerate parabolic problems (UMA).
Efficient methods for the numerical solution the Schrödinger equation in the semiclassical limit were analysed. At the atomistic-continuum coupling, a new error analysis of coupled methods has been developed (UOS, FORTH). In addition, the effect of multi-body potentials to the analysis has led to preliminary results.
Research Theme 4: Applications.
Significant progress has been made on the development new shallow water type models their applications to describe precipitation and infiltration phenomena (UNIVAQ, UOS, Ambiental). The UMA team introduced coupled model for suspended and bedload sediment transport has been proposed in the shallow water framework. On another front UNIVAQ and UOS teams focused on fluid dynamical models for the polluted atmosphere. In order to tackle challenging computations in Astrophysics a novel second order accurate Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume scheme for nonlinear hyperbolic systems has been developed (UMA). In materials and biomaterials, progress has been made at several fronts. UNIVAQ focused on models for polymeric fluids. In a collaborative effort UOS, UNIVAQ and KAUST studied models for viscoelastic materials. New mathematical techniques were introduced to study the mechanical behaviour of the extracellular matrix (ECM) caused by cell contraction (FORTH, UOS). FORTH & UOS teams introduced and studied appropriate numerical approximations used very successfully in the computational experiments.
