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Center for Nonlinear Studies |
Discrete calculus methods are a numerical approach to the solution of partial differential equations wherein calculus is discretized exactly. As a result, the discrete divergence, gradient, and curl operators are exact and therefore mimetic. All the numerical error in these methods occurs within the approximation of the material constitutive equations. In this talk, it is shown how the discrete calculus approach can be generalized to construct higher-order methods. The approach is quite general, but for clarity we demonstrate how a third-order discretization of the unsteady diffusion equation would be constructed. Higher-order discrete calculus methods are interesting because they have more unknowns per cell and a mathematical structure similar to a finite element method, but have the conservation properties and general flavor of a finite volume method. The presented method is exact for piecewise quadratic solutions and shows third-order convergence on arbitrary triangular/tetrahedral meshes. The numerical accuracy of the method is confirmed on both two-dimensional and three-dimensional unstructured meshes. The computational cost required for a desired accuracy is analyzed against first-and second- order discrete calculus and finite volume methods.
Host: Mikhail Shashkov