We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7: 100066, 2020] using the random batch strategy. The collisions only take place inside the small but randomly selected batches so that the computational cost is reduced to O(N ) per time step. Meanwhile, our methods can preserve the conservation of mass, momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods.

Off-site construction has been increasingly employed due to its advantages, for instance, improved quality control, reduced skills labour, faster construction time, decreased material wastage and safe working environment. As the most cutting-edge off-site construction, modular buildings have been utilised for residential building, student accommodation, and hotel projects. However, because of existing and underlying constraints, the adoption of modular buildings is still relatively low. To reveal factors hindering the development of high-rise modular buildings, a comprehensive literature review, coupled with a focus group study, were conducted. A questionnaire survey inquiring about all stakeholders was implemented to quantify constraints. The results were further examined according to a real-life case study. This paper manifested that “Lack of coordination and communication among stakeholders”, “Higher cost”, “Lack of government support”, “Lack of experience and expertise”, “Lack of building codes and standards”, “Poor supply chain integration”, and “Complexity of connection” are the top barriers curbing the uptake of modular buildings. The findings should provide a valuable reference for stakeholders adopting modular buildings, whilst mitigating risks amid modular construction. Future research is expected to exploit building information modelling and design for manufacture and assembly to alleviate these existing constraints and promote the performance of modular construction as well.

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We seek to establish qualitative convergence results to a general class of evolution PDEs described by gradient flows in optimal transportation distances. These qualitative convergence results come from dynamical systems under the general name of LaSalle Invariance Principle. By combining some of the basic notions of gradient flow theory and dynamical systems, we are able to reproduce this invariance principle in the setting of evolution PDEs under general assumptions. We apply this abstract theory to a non-exhaustive list of examples that recover, simplify, and even extend the results in their respective literatures.

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