The goal of this project is to provide a new state-of-the-art Numerical Methods and High Performance Computing (HPC) software for the numerical simulation of Bose-Einstein condensates (BEC). This is a timely objective, since BEC physics is a very dynamic research field, with applications belonging to a future technological era. The project bridges a gap in this field, where modern, HPC numerical codes are nowadays absent. With the purpose to develop robust and reliable numerical simulators, based upon new mathematically sound methods and modern HPC strategies, this project has no worldwide equivalent and will strongly impact studies of BEC physics conducted in both mathematics and physics communities. The project combines mathematical modelling, numerical analysis and simulation in a coherent workflow that brings together 20 (permanent) mathematicians and computer scientists from 4 partners. This includes a solid task-force made of 5 research engineers, who will use their important experience in HPC to support coding effort. The project will also take benefit from the strong interaction with external collaborators, who are expert physicists in BEC systems. This participation ensures the mandatory critical mass required to take up the following challenges: (i) develop new high-order numerical methods with firm mathematical background; (ii) develop an integrated and resilient open-source HPC software that will materialize advances in numerical methods and algorithms for BEC simulation; (iii) apply these codes to numerically reproduce realistic physical configurations that are not possible to simulate with presently existing software. With regard to these objectives, the project fits the call of Numerical Models ANR program, action Basic Research in Modelling and Simulation of Complex Systems, with the purpose to "understand and predict" complex physical phenomena. To cover all these challenging topics, the project is divided in 6 tasks with a total number of 26 subtasks. Each task involves at least two partners, and often three to four partners, to encourage exchanges and communication. Most of the subtasks are novelties in the BEC research field and some of the subtasks are highly challenging (e.g. modeling of stochastic effects, high order methods in HPC codes, simulation of real experiments, etc). The team will be strengthened with 2 PhD students and 5 postdoctoral students. Theoretical results will be disseminated through first rank publications, while developed HPC codes will be made available from a dedicated open-access Web-site. This will allow researchers to adapt the codes for their own purposes and thus maintain competition in this very dynamic field. The project will also be a unique opportunity to set a new vivid community of research in the emerging field of the numerical simulation of BEC systems. In order to ensure the training of new researchers and the resilient character of the developed software, two workshops and a summer school will be organized in the framework of the project.