Petrov–Galerkin flux upwinding for mixed mimetic spectral elements, and its application to geophysical flow problems

A comparison of variational upwinding schemes for geophysical fluids, and their application to potential enstrophy conserving discretisations

“…For the sake of brevity we introduce the inner product notation as a, b = abdΩ for scalar fields a, b and v, u = v • udΩ for vector fields. Following (8), the variational derivative of the potential vorticity q h with respect to the velocity u h is given as the matrix…”

“…In the present article the application of the APVM and SUPG methods are compared to a more recent approach by which the potential vorticity trial functions are evaluated at downstream locations within the reference element [8] in terms of their consistency, conservation properties and residual errors. This reference element stabilisation has been used previously for the stabilisation of both the potential enstrophy for the shallow water equations, in which case it is applied to the downwinding of the trial functions [8], and to the potential temperature in the 3D compressible Euler equations, in which case it is applied as the upwinding of the test functions [9].…”

“…In the present article the application of the APVM and SUPG methods are compared to a more recent approach by which the potential vorticity trial functions are evaluated at downstream locations within the reference element [8] in terms of their consistency, conservation properties and residual errors. This reference element stabilisation has been used previously for the stabilisation of both the potential enstrophy for the shallow water equations, in which case it is applied to the downwinding of the trial functions [8], and to the potential temperature in the 3D compressible Euler equations, in which case it is applied as the upwinding of the test functions [9]. In the context of strong form differential operators, for which the differential operator is applied to the trial function (such as the divergence of a mass flux in H(div, Ω) for the flux form advection of some quantity in L 2 (Ω)), this is achieved via the upwinding of the test function (provided that the test function is C 0 continuous across element boundaries in the direction of the flow).…”

“…In the context of strong form differential operators, for which the differential operator is applied to the trial function (such as the divergence of a mass flux in H(div, Ω) for the flux form advection of some quantity in L 2 (Ω)), this is achieved via the upwinding of the test function (provided that the test function is C 0 continuous across element boundaries in the direction of the flow). For a weak form differential operator however (such as the material advection of the same variable via the same mass flux) this reference element upwinding is conversely applied via the downwinding of the trial functions themselves [8].…”

Energy conserving SUPG methods for compatible finite element schemes in numerical weather prediction

“…Within the Poisson bracket framework, an energy conserving SUPG method was considered for the evolution of potential vorticity in the shallow water equations in [4], exploiting the fact that the Poisson bracket term corresponding to the curl part of the velocity transport term's vector-invariant form is antisymmetric in itself. Another Petrov Galerkin type method exploiting the latter fact is given in [21], where the potential vorticity is diagnosed using Lagrangian trial functions, which are evaluated at downstream locations.…”