One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG’s intrinsic (upwind) dissipation being biased towards high
frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where
limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial
diffusion, such as spectral vanishing viscosity (SVV). The latter, although recognised as very useful in
CG-based high-fidelity turbulence simulations, has been observed to be sub-optimal when compared to
DG at intermediate polynomials orders (P ≈ 3). In this paper we explore an alternative stabilisation
approach by the introduction of a continuous interior penalty on the gradient discontinuity at elemental
boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this
introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of
SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior)
to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious
choice of GJP’s P-dependent scaling parameter is made and found to be consistent with previous apriori error analysis. The favourable properties of the GJP stabilisation approach are also supported by
turbulent flow simulations of the incompressible Navier-Stokes equation, as we achieve high-quality flow
solutions at P = 3 using GJP, whereas SVV performs marginally worse at P = 5 with twice as many
degrees of freedom in total
