Introduction, objectives and overview of the research programme
This network is focused on the training of young researchers (ESRs) in the general area of nonlinear hyperbolic and convection dominated PDEs (HCD-PDEs) with emphasis on innovative modelling and computational methods. Computational and Applied Mathematics is becoming one of the main driving forces for the current development of different fields of science ranging from physics and engineering to biology and medicine. Modelling, Analysis and large-scale Computer Simulations based on advanced mathematical methods are absolutely essential in modern science and engineering. The research program within this network will be centered in a prominent (in terms of both history and importance) field, that is placed at the very forefront of modern Computational and Applied Mathematics. In fact, HCD-PDEs, is one of the few fields of applied mathematics where
- models originate from the need to describe physical phenomena of great importance;
- a plethora of highly influential analytical methods has been developed over the years to study these models;
- pioneering and advanced computational methods were designed to approximate HCD-PDEs, particularly to approximate phenomena of interest such as shocks, phase transitions, propagating interfaces etc.
The fact that hyperbolic convection dominated PDEs is one of the few areas within Computational and Applied Mathematics, where traditionally modelling, Physics, Mechanics, analytical approaches, and advanced compu- tational methods are linked, and have contributed in synergy to several achievements till date, makes this field eminently suitable to train young researchers in. These researchers can become research leaders in a wide area as well as impact both industry and non-academic scientific institutions.
The network consists of some of Europe’s leading research groups on hyperbolic PDEs, including experts on Modelling, Analysis and Computation. Although the emphasis will be the subtle modelling of shocks, interfaces and other singular phenomena related to such PDEs and their robust and efficient computation, the relevant analysis issues of these PDEs will also be a main task of this network. The innovative techniques developed will be applied to diverse concrete problems ranging from fluid dynamics and geophysical flows to materials science. The Research projects of the proposed training programme are designed in order to address a number of challenges in the field which constitute an exciting research programme. The research projects (PhD Projects) are divided into four thematic packages:
• Research Theme 1: Measure Valued Solutions and Uncertainty Quantification
• Research Theme 2: Propagating Interfaces
• Research Theme 3: Models and Methods across scales
• Research Theme 4: Applications.
Overall ModCompShock is designed around a balanced distribution of research projects within the training program with respect to:
- involvement of the participants in various training functions;
- responsibility of participants on specific research programmes and secondment distribution;
- complementarity of skills and expertise with respect to analysis, modelling and computation, as well as with respect to diverse methodological approaches;
- essential contribution of Partner organisations by complementing missing expertise in research and trans- ferable skills.
Overview and content structure of the training
In Europe today there are several approaches to PhD education in Computational and Applied Mathematics. Due to the very high scientific level of researchers active in the field and its importance to technological development, these are quite effective albeit scattered. We believe that an intense, systematic and focused training at a European level has much to offer in terms of both increasing the capacity and of promoting excellence of the overall training. Our approach is based both on structural and collaborative forms of PhD training, increasing significantly the capacity found locally and promoting excellent research in the area of Computational and Applied Mathematics at a European level for the benefit of the trained fellows.
Given our research priorities described in Section 2.1, our approach to training is based on developing, in a systematic way, the skills of early stage researchers (ESRs) creating a modern research profile in Computational and Applied Mathematics based on:
- Deep scientific knowledge and ability to use computational, analytical and modelling techniques.
- Diversity in both topics and methodologies and exposure to diverse scientific points of view; In the proposed area of research this can only be attained at a European level.
- Real capacity for interdisciplinary research. The design of the training program needs to, first, be structured and monitored in a way that fosters the devel- opment of these traits, and, second, to be adapted to the skill set and educational needs of the individual ESR. The training program, involves several actions and activities which are targeted to contribute to: • excellent PhD theses
• to enhance their important qualitative characteristics as computational and applied mathematicians, • to endow the researchers with transferable skills for successful career development. In particular the training program aims for ESRs to develop the following skills,
- Analytical thought and efficient use of analytic techniques
- Computational competence through training in developing efficient numerical algorithms and their software implementation
- modelling skills as well as the ability to communicate with other applied scientists.
- Inter-European mobility and the ability to work with multiple advisors and in different research environments. The proposed training program and the included research projects are designed to enhance these qualitative characteristics.