Is there a software-engineering methodology for functional programming?
Personally I find that all the usual good practices from OO development apply in functional programming as well - just with a few minor tweaks to take account of the functional worldview. From a methodology perspective, you don't really need to do anything fundamentally different.
My experience comes from having moved from Java to Clojure in recent years.
Some examples:
Understand your business domain / data model - equally important whether you are going to design an object model or create a functional data structure with nested maps. In some ways, FP can be easier because it encourages you to think about data model separately from functions / processes but you still have to do both.
Service orientation in design - actually works very well from a FP perspective, since a typical service is really just a function with some side effects. I think that the "bottom up" view of software development sometimes espoused in the Lisp world is actually just good service-oriented API design principles in another guise.
Test Driven Development - works well in FP languages, in fact sometimes even better because pure functions lend themselves extremely well to writing clear, repeatable tests without any need for setting up a stateful environment. You might also want to build separate tests to check data integrity (e.g. does this map have all the keys in it that I expect, to balance the fact that in an OO language the class definition would enforce this for you at compile time).
Prototying / iteration - works just as well with FP. You might even be able to prototype live with users if you get very extremely good at building tools / DSL and using them at the REPL.
For Clojure, I recommend going back to good old relational modeling. Out of the Tarpit is an inspirational read.
Thank God that the software-engineering people have not yet discovered functional programming. Here are some parallels:
Many OO "design patterns" are captured as higher-order functions. For example, the Visitor pattern is known in the functional world as a "fold" (or if you are a pointy-headed theorist, a "catamorphism"). In functional languages, data types are mostly trees or tuples, and every tree type has a natural catamorphism associated with it.
These higher-order functions often come with certain laws of programming, aka "free theorems".
Functional programmers use diagrams much less heavily than OO programmers. Much of what is expressed in OO diagrams is instead expressed in types, or in "signatures", which you should think of as "module types". Haskell also has "type classes", which is a bit like an interface type.
Those functional programmers who use types generally think that "once you get the types right; the code practically writes itself."
Not all functional languages use explicit types, but the How To Design Programs book, an excellent book for learning Scheme/Lisp/Clojure, relies heavily on "data descriptions", which are closely related to types.
So what is the methodology for a systematic (model-based ?) design of a functional application, i.e. in Lisp or Clojure?
Any design method based on data abstraction works well. I happen to think that this is easier when the language has explicit types, but it works even without. A good book about design methods for abstract data types, which is easily adapted to functional programming, is Abstraction and Specification in Program Development by Barbara Liskov and John Guttag, the first edition. Liskov won the Turing award in part for that work.
Another design methodology that is unique to Lisp is to decide what language extensions would be useful in the problem domain in which you are working, and then use hygienic macros to add these constructs to your language. A good place to read about this kind of design is Matthew Flatt's article Creating Languages in Racket. The article may be behind a paywall. You can also find more general material on this kind of design by searching for the term "domain-specific embedded language"; for particular advice and examples beyond what Matthew Flatt covers, I would probably start with Graham's On Lisp or perhaps ANSI Common Lisp.
What are the common steps, what artifacts do I use?
Common steps:
Identify the data in your program and the operations on it, and define an abstract data type representing this data.
Identify common actions or patterns of computation, and express them as higher-order functions or macros. Expect to take this step as part of refactoring.
If you're using a typed functional language, use the type checker early and often. If you're using Lisp or Clojure, the best practice is to write function contracts first including unit tests—it's test-driven development to the max. And you will want to use whatever version of QuickCheck has been ported to your platform, which in your case looks like it's called ClojureCheck. It's an extremely powerful library for constructing random tests of code that uses higher-order functions.