Discuss the various methodologies employed in the design and analysis of sustainable urban drainage systems $($SUDS$)$ in civil engineering. Explore the hydrological and environmental considerations that must be taken into account, as well as the integration of green infrastructure elements such as permeable pavements, green roofs, and bio$-$retention cells. Evaluate the potential challenges in the implementation of SUDS in densely populated urban areas, considering factors like land availability, cost, and maintenance requirements. Additionally, address the role of advanced modeling software in predicting the performance of these systems under varying climatic conditions, and how these tools can be used to optimize the design for maximum efficiency and sustainability.