title={Sub-cell shock capturing for {D}iscontinuous {G}alerkin methods},

author={Persson, P.-O. and Peraire, J.},

booktitle={44th AIAA Aerospace Sciences Meeting and Exhibit},

pages={112},

year={2006}

}

@inbook{MenDeG14,

Annote = {doi:10.2514/6.2014-2923},

Author = {Gianmarco Mengaldo and Daniele De Grazia and Freddie Witherden and Antony Farrington and Peter Vincent and Spencer Sherwin and Joaquim Peiro},

@@ -444,22 +444,39 @@ Under the two following sections it is possible to define the initial conditions

\section{Examples}

\subsection{Shock capturing}

Compressible flow is characterised by abrupt changes in density within the flow domain often referred to as shocks. These discontinuities lead to numerical instabilities (Gibbs phenomena). This problem is prevented by locally adding a diffusion term to the equations to damp the numerical fluctuations. These fluctuations in an element are identified using a sensor algorithm which quantifies the smoothness of the solution within an element. The value of the sensor in an element is defined as

Compressible flows can be characterised by abrupt changes in density within the flow domain often referred to as shocks. These discontinuities lead to numerical instabilities (Gibbs phenomena). This problem is prevented by locally adding a diffusion term to the equations to damp the numerical oscillations.

An artificial diffusion term is introduced locally to the Euler equations to deal with flow discontinuity and the consequential numerical oscillations. Two models are implemented, a non-smooth and a smooth artificial viscosity model.

For the non-smooth artificial viscosity model the added artificial viscosity is constant in each element and discontinuous between the elements. The Euler system is augmented by an added laplacian term on right hand side of equation \ref{eq:euler}. The diffusivity of the system is controlled by a variable viscosity coefficient $\epsilon$. The value of $\epsilon$ is dependent on $\epsilon_0$, which is the maximum viscosity that is dependent on the polynomial order ($p$), the mesh size ($h$) and the maximum wave speed and the local sensor value. Based on pre-defined sensor threshold values, the variable viscosity is set accordingly

For the non-smooth artificial viscosity model the added artificial viscosity is constant in each element and discontinuous between the elements. The Euler system is augmented by an added Laplacian term on right hand side of equation \ref{eq:euler}\cite{persson2006sub}.

The diffusivity of the system is controlled by a variable viscosity coefficient $\varepsilon$.

For consistency $\varepsilon$ is proportional to the element size and inversely proportional to the polynomial order.

Finally, from physical considerations $\varepsilon$ needs to be proportional to the maximum characteristic speed of the problem.

where $\langle\cdot, \cdot\rangle$ represents a $L^2$ inner product, $q$ and $\tilde{q}$ are the full and truncated expansions of a state variable (in our case density)

\caption{(a) Steady state solution for $M=0.8$ flow at $\alpha=1.25^\circ$ past a NACA 0012 profile, (b) Artificial viscosity ($\epsilon$) distribution}

\caption{(a) Steady state solution for $M=0.8$ flow at $\alpha=1.25^\circ$ past a NACA 0012 profile, (b) Artificial viscosity ($\varepsilon$) distribution}