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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - ModCompShock (Modelling and Computation of Shocks and Interfaces)

Teaser

Summary

The ITN entitled Modelling and Computation of Shocks and Interfaces (ModCompShock) is focused on the training of young researchers (ESRs) in the general area of nonlinear hyperbolic and convection dominated Partial Differential Equations (HCD-PDEs), with emphasis on innovative modelling and computational methods.

The area of Computational and Applied Mathematics on which this project is focused is a fundamental scientific field for the development of a wide range of key technologies. Predictive science ranging from Geophysics, to Biology, Medicine to Material design, rely on modern Computational and Applied mathematics. These are related to European priority areas and have strong connections to various branches of European high-tech industry.

The synergetic interplay of modelling, analysis and large-scale computer simulations based on advanced mathematical methods plays a prominent role in the present programme. The ERSs are being trained in a wide inter-disciplinary area and can become research leaders as well as impact both industry and non-academic scientific institutions. In the pursuance of this goal, the research groups are assisted by experts in these areas of application and non-academic partners, resulting in a significant enhancement of the impact of the research and training.
The Research projects of the training programme are designed in order to address a number of challenges in the field which constitute an exciting research programme. The research projects are divided into four thematic packages:

â€¢ Research Theme 1: Measure Valued Solutions and Uncertainty Quantification

â€¢ Research Theme 2: Propagating Interfaces

â€¢ Research Theme 3: Models and Methods across scales

â€¢ Research Theme 4: Applications.

Work performed

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 was resolved (Lâ€™Aquila and Sussex). The structure and properties of the equations of polyconvex thermoelasticity is the subject of a study by FORTH and KAUST. Issues related to ergodicity of spherically symmetric fluid flows related to black holes are addressed (Paris) New computational methods were proposed and studied. First, Multi-Level Monte Carlo method: a paradigm for UQ in tsunami simulations has been proposed by the Malaga and ETH team. Fully Discrete Approximation of Parametric and Stochastic Elliptic PDEs (Paris and ETH) were designed and analysed.

Research Theme 2: Propagating Interfaces

The Lâ€™Aquila team addressed issues related to the existence and regularity of solutions and scale limit analysis. Significant progress has been made in the study of the quantitative rigidity of almost umbilical hypersurfaces (Zurich). The Paris 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, Malaga, Paris),computational methods of non conservative products using Roe-type path conservative scheme (Paris). The Oslo team studied numerical methods for the Kuramoto-Sakaguchi equations describing the behaviour of weakly coupled oscillators. Further, the application of a spectral method to the Euler equations was studied (ETH,Oslo).

Research Theme 3: Models and Methods across scales

The Aachen team is working on the development of a novel Lagrangian method for the simulation of compressible fluid flows. Towards the development of adaptive Galerkin methods for nonlinear hyperbolic problems, a derivation of a stable variational formulation for scalar nonlinear conservation laws has been proposed (Aachen).
A very promising approach for data assimilation and reduced modeling was initiated (Aachen, Paris). A new formulation leading to entropy conservative and entropy diminishing schemes was introduced (Sussex). The framework of entropy-stability introduced by Tadmor for systems of conservation laws has been applied to develop entropy-stable methods for degenerate parabolic problems (Malaga). At the atomistic-continuum coupling, a new error analysis of coupled methods has been developed (Sussex, FORTH).

Research Theme 4: Applications.

It has been made on the development new shallow water type models to describe precipitation and infiltration phenomena (Lâ€™Aquila, Sussex, Ambiental). The groups of Catania, Malaga, Paris, have contracted to the design of efficient schemes for shallow water type equations at several fronts. Application on fluid dynamical models for the equations governing the motion of the atmosphere was studied (Lâ€™Aquila). Preliminary results on the study of modeling and simulation of powder cloud avalanches, which are identified as a major environmental hazard in montane regions are available by the ETH team. Regarding, materials and bio-materials, new mathematical techniques were introduced to study the mechanical behaviour of the extracellular matrix (ECM) caused by cell contraction (FORTH, Sussex).

Final results

It is expected that significant progress until the completion of this project will be done mainly at the following scientific challenges:

It is expected that new Computational methods for measure valued solutions and Uncertainty Quantification (UQ) and stochastic HCD-PDEs will be developed. Important progress is expected for problems related to propagating phase boundaries and small scale dependent shock waves. Furthermore, we shall study Dispersive Shocks from various perspectives. Such shock waves where the dominant regularization mechanism is a small dispersion, have been realised experimentally in many diverse systems including plasmas, water waves, Bose-Einstein condensation (BEC) and nonlinear optics. A key objective of our project is to have significant advances related to Methods and Models across scales. Most realistic models in Physics and Engineering involve a large range of spatiotemporal scales. We consider methods that have the ability to be adapted to the singular and/or complex character of the desired solutions. Our research will have impact on new developments is several concrete real world problems related to Aerodynamics design of new Materials and Nano-Materials. Study of Bio mechanical properties in BioMedical applications, Geophysical flows and other problems.

Technology and the advent of computing power has dramatically altered scientific research and development over the last two decades. Innovation relies to a large extent on accurate prediction and computation. The training program and the synergetic effect of this field of study will guarantee that our fellows will have the precise skills needed for the next generation of high-level scientists with mathematical education, namely; analytical thought, computational competence, and modelling skills as well as the ability to communicate with scientists and engineers. As mathematical modelling is absolutely essential in modern predictive science, our fellows will have the training and the skills for successful careers in both the Academic as well as the non-Academic sector.