Which methodology of a research paper around a proposed software system
What you've described is called design science or design science research; it is most certainly not a case study. In a comment on your original post, @XLeitix nailed it as the pattern: 


  problem -> proposed solution -> evaluation [rinse and repeat until results are ok]


In conducting a broad-based literature review of research on Wikipedia, my colleagues and I had to come up with a list of different research designs. Design science is in a category of its own, because it is quite different from the methods in social sciences. Design science is the default approach to research in computer science and software engineering, and so no one is going to explain the "research method" when they follow this pattern. It arises from the engineering paradigm where the goal of research is not to discover what exists or how or why things work, but rather to create something that is useful to people and prove that it is useful (usually by comparing its usefulness to some specified benchmark).

Ironically, the name "design science" does not come from computer science or software engineering, but rather from information systems (IS) research. IS research is primarily social science research, and so IS researchers had to come up with a name or a way to describe this type of research that is so different from social science, and yet scientifically rigorous. There are at least two major articles that describe its scientific characteristics in depth:


von Alan, R. Hevner, et al. "Design science in information systems research." MIS quarterly 28.1 (2004): 75-105.
Gregor, Shirley, and David Jones. "The anatomy of a design theory." Journal of the Association for Information Systems 8.5 (2007): 312.