Purely "discrete" PDEs?

Question

Usually, one formulates a system of continuous PDEs and then discretizes it in order to approximately solve it.

Is there a view point that instead formulates a system of "discrete" PDEs, which therefore do not require a discretization step in order to solve it, even if some other type of reformulation may be required?

Answer 1

You might be interested in the area of Discrete Differential Geometry.

The behavior of physical systems is typically described by a set of continuous equations using tools such as geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation, one must derive discrete (in space and time) representations of the underlying equations. Researchers in a variety of areas have discovered that theories, which are discrete from the start and have key geometric properties built into their discrete description, can often more readily yield robust numerical simulations that are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm.

Answer 2

You can consider cellular automata. See Cellular Automata Modeling of Physical Systems (Chopard & Droz, CUP 1998) for the application of cellular automata to modeling physical systems.