Discrete calculus methods for diffusion

A general methodology for the solution of partial differential equations is described in which the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well. The construction of both node and cell based methods of first and second order are described for the problem of unsteady heat conduction though the method is applicable to any PDE system. The performance of these new methods are compared to classic solution methods on unstructured 2D and 3D meshes for a variety of simple and complex test cases.