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In last decades, the interest in Discontinuous Galerkin (DG) methods for conservationlaws and convection-dominated problems has continuously increased. Their suitability

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to construct robust stabilized high-order numerical schemes on arbitrary unstructuredand non-conforming grids has been proved for a variety of physical phenomena, see, for

instance, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Among all DG methods, the Hybridizable

Discontinuous Galerkin (HDG) method outstands for implicit schemes, mainly

due to its reduced number of degrees of freedom, and its superconvergence properties

14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25.

However, the inherent stability of DG methods is not enough to overcome local numerical

oscillations in the vicinity of sharp fronts, for high-order computations. The numerical

perturbations deteriorate the solution, and may preclude convergence for non-linear problems.

As a remedy, many researchers have extended classical shock-capturing methodologies

from nite dierences (FD) and nite volume (FV) schemes to high-order DG methods

27, 28, 29, 30, 31, 32. A widely used strategy to capture shocks is articial diusion or viscosity.

The idea was rst introduced in 35 and further developed in 36, 37, and it became

very popular with the Streamline Upwind Petrov-Galerkin (SUPG) nite element methods,

see 38, 39, 40, 41. In the last decades, articial viscosity has also been applied to diminish

numerical oscillations in DG computations, see for instance 42, 5, 7, 4, 43, 44, 45. As explained

in 31, articial viscosity expands the thickness of the shock layer so that it safely

exceeds the resolution length scales of the numerical method and eliminates the spurious

oscillations”. Hence, the amount of articial viscosity is of great importance and some authors

propose a sub-cell based articial viscosity to improve the accuracy of the solution in

the vicinity of shocks, introducing less articial viscosity, see for instance 28, 46. Articial

diusion techniques can capture shocks in a robust and accurate manner, however, determining

the suitable amount of articial viscosity is not straightforward. On other hand, the

Total Variation Diminishing methods bound the variations in the numerical solution, so

that no new local extrema forms in the domain, see for instance 47. These methods were

initially designed in the context of FD and FV 48, 49, 50, 51, and the idea has also been

developed in the Runge-Kutta Discontinuous Galerkin (RKDG) methods 52, 53, 54, 55.

RKDG methods are specially designed to damp spurious oscillations, but the order of the

approximation is reduced to linear or constant in the vicinity of shocks, and accuracy can

only be improved by mesh renement. In a dierent approach, Huerta et al. 56 propose

taking prot of the stabilization induced by the numerical

uxes to capture sharp fronts

of the solution inside high-order elements for the solution of the Euler equations of gas

dynamics, with the Local Discontinuous Galerkin (LDG) method. Sub-cells are considered

inside stabilized elements, in the vicinity of shocks, to dene an elemental discontinuous

approximation space, and numerical

uxes across sub-cell boundaries are introduced. By

means of this shock-capturing technique, the order of the approximation is reduced only

in the elements where the solution is not smooth. Thus, the high-order accuracy of order

p + 1, in the large majority of the domain, is locally decreased to order h=p only in the

elements where the shock is contained, being p the degree of approximation, and h the

element size.

This paper proposes a novel shock-capturing technique for high-order HDG methods,

with application to convection-diusion and compressible Navier-Stokes equations. Following

the idea in 56, the strategy is to exploit the stabilization induced by DG numerical

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fluxes to capture sharp fronts of the solution inside high-order HDG elements. First, adiscontinuity sensor developed in 28 is used to detect the elements aected by sharpfronts. This discontinuity sensor is based on the rate of decay of the coecients of theapproximated solution, and quanties the smoothness of the solution with an elementalscalar. Based on the smoothness of the solution, the approximation space inside stabilizedelements and faces is modied, from a standard continuous representation of the solutionto a piecewise constant approximation. The basis of discontinuous shape functions insidethe stabilized elements and faces is produced by using non-overlapping sub-cells, such thateach sub-cell contains one node. The weak form corresponding to the HDG local problemis modied to take into account the discontinuities inside stabilized elements, introducingDG numerical uxes across sub-cells boundaries. As a result, the numerical uxes insideelements provide additional stabilization with no additional Degrees of Freedom (DOFs).By means of this shock-capturing technique, sharp fronts can be captured without modifyingthe DOFs or mesh topology, with a reduction of the order of approximation to orderh=p only in stabilized elements.The outline of this paper is as follows. The continuous and discontinuous nodal basisfor standard and stabilized elements are presented in section 2: the elemental discontinuousspace proposed in 56 is considered for the approximation in elements, and theapproximation space on stabilized faces is also dened. The shock sensor proposed in 28is recalled in section 2.1. Section 3 presents the stabilized HDG method for convectiondiusion problems, with a modied weak form for the local problem in stabilized elements;DG numerical uxes are introduced on the boundary of the sub-cells to account for the discontinuousapproximation in stabilized elements, providing additional stabilization to theHDG formulation. Following the same ideas, the stabilized HDG formulation for the compressibleNavier-Stokes equations is presented in Section 4. Finally, numerical examplesin section 5 show the excellent performance of the proposed shock-capturing technique indiminishing numerical oscillations is the vicinity of sharp fronts, for the high-order solutionof two dimensional steady convection-diusion and compressible Navier-Stokes equations.