Discrete calculus methods and their implementation
The Discrete Calculus (DC) Approach is introduced as a general methodology for the solution of partial differential equations (PDEs). In this approach, the discretization of the calculus is exact and all approximation occurs as an interpolation problem on the material constitutive equations. Numerical methods, whose underlying discretization is exact, are classified as Discrete Calculus Methods. The fact that the calculus is exact gives these methods the ability to capture the physics of PDE systems well.
Some existing discretization methods with desirable mathematical and physical properties are derived using the Discrete Calculus methodology. While these methods were first derived and later shown to possess attractive properties, the Discrete Calculus Approach is used to show how these and other novel Discrete Calculus schemes can be derived from the outset. In particular, the construction of node-based, cell-based and face-based DC schemes of first and second order are described for the problem of unsteady heat conduction—though the DC methodology is applicable to any PDE system. In addition, a novel third order Discrete Calculus method is derived in order to illustrate the construction of higher order DC methods. 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.
A generic, object-oriented, parallel, three-dimensional, unstructured software code has been developed to implement these methods and the software implementation of these methods is described. Parallelization of the 3D code is shown to achieve a high computational efficiency. This code structure and the Discrete Calculus operators are then used to implement the classical and the exact fractional step methods for discretizing the incompressible Navier-Stokes equations.