Abstract

The development of various numerical methods capable of accurately simulating fluid flow has evolved greatly over
time. In the past years, discontinuous Galerkin methods have seen great interest for problems such as Large Eddy
Simulations, Aeroacoustics, incompressible and even compressible flows. These methods attractiveness some from their
ability to easily increase the order of accuracy thus yielding more precise solutions. These methods use higher order
polynomials, which can easily be increased or decreased within the element, while allowing for discontinuities between
elements. When shocks and discontinuities are present in a simulation, particular attention must be taken to avoid Gibbs
phenomenon within the elements. This phenomenon occurs when steep gradients in the solution are present causing
the solution to have erratic oscillations typically associated with the higher order terms of the integrating polynomial.
These oscillations in turn lead to non-physical solutions such as negative pressures and therefore need to be controlled.
A variety of methods have been developed to mitigate the oscillatory behavior of discontinuous Galerkin methods when
steep gradients are present, a very promising method is the addition of artificial viscosity in order to diminish the effects
of the non-physical oscillations. Adding a viscous term to the conservation equations being solved can inevitably lead to
inaccurate solutions if it is added in excessive amounts. The balance between damping of the non-physical oscillations
and minimizing the amount of artificial viscosity added can if the location of the shocks in the flow field is known.
This intricate balance is achieved by ensuring that the functions used to find the areas of concern are not overlapping
shock regions with smooth regions and when viscosity is added it is important that it is limited to ensure that it will not
completely dissipate the real solution.