  We present Decapodes, a diagrammatic tool for representing, composing, and
solving partial differential equations. Decapodes provides an intuitive
diagrammatic representation of the relationships between variables in a system
of equations, a method for composing systems of partial differential equations
using an operad of wiring diagrams, and an algorithm for deriving solvers using
hypergraphs and string diagrams. The string diagrams are in turn compiled into
executable programs using the techniques of categorical data migration, graph
traversal, and the discrete exterior calculus. The generated solvers produce
numerical solutions consistent with state-of-the-art open source tools as
demonstrated by benchmark comparisons with SU2. These numerical experiments
demonstrate the feasibility of this approach to multiphysics simulation and
identify areas requiring further development.
