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# Discrete differential geometry

Partial differential equations are equations with respect to derivatives of some unknown functions. To solve them numerically one might "sample" or "discretize" these derivatives via finite differences. Instead, we look at the derivations of of some typical PDEs (heat, wave, fluid flow etc) and, based on the physics, represent all the quantities involved as differential forms or, better, *discrete* differential forms whenever possible. These quantities have to satisfy the basic laws of physics (conservation of energy, mass, etc) that take a form of differential and algebraic equations. The discrete versions of these equations provide ready-made simulations (i.e., numerical solutions) of these PDEs. They are similar to cellular automata. The advantage of this approach is that the laws of physics, by design, are satisfied *exactly* rather than approximately, as is the case with discretization of PDEs. The next main issue is that, even if the discrete PDE is physically valid, it may not approximate its continuous counterpart. This fact is frequently revealed by the former's anisotropic behavior. We follow a broad approach to these issues by developing, in addition to differential calculus, its discrete version as well as other possible calculi. We borrow the necessary tools from algebraic topology.

- The algebra of chains
- Cell complexes and simplicial complexes
- Manifolds
- Homology as a vector space
- Boundary operator
- Properties of homology groups
- Cell maps, simplicial maps and their homology maps

- Calculus of discrete differential forms
- Discretization of calculus
- Calculus is topology
- Cubical complexes
- Isotropy in numerical PDEs
- Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni

Simulations of PDEs based on the physics

- wave equation,
- fluid flow produced by a vector field and/or particle modeling,
- lattice gas cellular automata,
- heat transfer and diffusion, modeling with discrete exterior calculus;
- Maxwell equations.

Tools of discrete exterior calculus: