Structure preserving discretisations: from structure to results
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Structure-preserving discretisations, by construction, enable the conservation of many invariants of the dynamics of PDEs. For example, energy, momentum, helicity (or enstrophy in 2D), in the Navier-Stokes equations, also potential enstrophy in the shallow waters equations, magnetic and cross helicity in MHD, etc. Besides the formal relevance, there are also concrete practical benefits of employing numerical discretisations that preserve some of the structure of the original equations. In this work we will discuss: (i) the fundamental requirements for a structure preserving discretisation, (ii) the importance of the choice of formulation in the resulting properties of the numerical model, (iii) compare the results from a structure preserving discretisation to a standard approach. Structure preserving discretisations fundamentally rely on a set of function spaces that constitute a discrete de Rham sequence. We will discuss the properties of such a sequence of spaces. We will also relate the lack of such a sequence in the discretisation to some some known numerical challenges, e.g., checkerboard artifacts, etc. Although a prerequisite, the proper choice of discrete function spaces is not sufficient to construct a discrete physical model that preserves the invariants of the original system. The choice of formulation used to represent the original system of PDEs is often one among many equivalent choices. Different formulations present different discrete properties (even when using the same set of function spaces) and thus the same discretisation. We will use the shallow waters equations and compressible Euler flow to show the relevance of deriving the PDE from a Hamltonian formulation. Finally, we will show results for the structure preserving discretisation of the shallow waters equations on the surface of a sphere and compare it to a standard (and widely used) approach.