Marc Gerritsma (TU Delft)

Structure preserving discretizations for computational physics

A physical model consists of conservation laws, equilibrium conditions, definitions and constitutive relations [1]. The conservation laws, equilibrium conditions and definitions are so-called topological relations. More specifically, these topological relations are independent of metric. The topological relations are divided into two disjoint classes which we refer to as outer-oriented relations and inner-oriented relations. Physically the model is closed by adding the material dependent constitutive relations. Mathematically, the model is closed by adding the metric-dependent constitutive relations which connect the inner-oriented topological relations to the outer-oriented topological relations. The accuracy of the physical model is determined by the accuracy of the constitutive relations.

Since the topological relations in a physical model are metric-free, we can represent them in a discrete setting without reference to mesh-dependent parameters. So these discrete topological relations hold on very coarse grids, but also on very fine grids in the limit when the mesh size goes to zero. Since these discrete relations hold universally they are called exact [2]. The inner-oriented relations are represented on one when mesh and the outer-oriented relations are represented on a dual mesh [3]. The metric-dependent constitutive relations do depend on both meshes and their mesh parameters. It is in these metric-dependent relations where the numerical error resides. The more accurately we can represent the constitutive relations, the more accurate our numerical scheme will be. So the constitutive relations play a special role in both physical and numerical modeling. These ideas will illustrated through a reaction-diffusion model for a fully conservative least-squares formulation, see [4].

References:
[1] E.Tonti, On the formal structure of physical theories, Prepr. Ital. Natl. Res. Counc.(1975).
[2] P. B. Bochev and J. M. Hyman, Principles of mimetic discretizations of differential equations, Compatible Spatial Discretizations
IMA Volumes in Mathematics and its Applications, 142,pp 89-119, 2006.
[3] J. Kreeft, A. Palha and M. Gerritsma, Mimetic framework on curvilinear quadrilaterals of arbitrary order, arXiv:1111.4304, 2011.
[4] P. Bochev and M. Gerritsma, A spectral mimetic least-squares method, submitted to CAMWA, 2014.