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?

score 3 · Accepted Answer · answered May 06 '14 at 07:14

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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.

answered May 06 '14 at 07:14

littleO

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I felt so giddy and excited reading that webpage. I wish I could have attended that conference. Anyway, I have this book on discrete and computational geometry (not discrete differential geometry) on my bookshelf -- should I start reading that as a pre-requisite? Or is discrete differential geometry pretty separate? – bzm3r May 08 '14 at 04:14
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It looks like really cool stuff! I haven't studied this material myself, but the course notes found here look like a good place to dive in. They might give you an idea of how much prerequisite knowledge from the textbook you mentioned is necessary. – littleO May 08 '14 at 11:29
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See also Peter Saveliev's site: Intelligent Perception, especially the material on discrete (exterior) calculus. As for Discrete and Computational Geometry by Devadoss & O'Rourke, even though I wouldn't call it a specific prerequisite for discrete differential geometry, it is a great introduction to important basic concepts such as polygons, polyhedra, triangulations and convex hulls. It even touches on discrete differential geometry in a section on the Gauss-Bonnet theorem for polyhedra. – J W Nov 30 '14 at 14:33

score 3 · Answer 2 · edited Dec 19 '14 at 23:38

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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.

edited Dec 19 '14 at 23:38

J W

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answered May 08 '14 at 19:13

user148606

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Purely "discrete" PDEs?

2 Answers2