Local Projection/ Limiting Techniques

Define the quantity $\Delta u$ at a new set of (three) collocation nodes $x_r$, $r=1,2,3$ inside a mesh element $i$.

$$\Delta u(x_r) = \left( \mathcal P(u_i)(x_r) - \overline u_i \right) \ ,$$ (25)

where the projection $\mathcal P$ may be the projection onto the space of linear function or the identity. If a cellwise neighborhood has been used in step 1 the nodes $x_r$ may be chosen at edge midpoints, whereas for the nodal neighborhood the nodes of the triangle are used.

If one insists that the reconstructed solution must not exceed the bounds established by the maximum and minimum volume averages of a chosen local neighborhood, denoted by $\Delta u^\mathrm{ref}$ the limiter will be activated whenever the magnitude of $\Delta u$ exceeds the magnitude of the allowed variation

$$\Delta u^{\mathrm{ref}} = \max(\overline u_i -u_{i,\min}, u_{i,\max} - \overline u_i)$$ (26)

at any of the nodes $x_r$. The quantity $\phi(x_r)$ may be defined as

$$\phi(x_r) =\left\{ \begin{array}{ll} 1 &\vert \Delta u(x_r) \vert... ...ef}} } {\vert \Delta u(x_r)\vert } & \mathrm{otherwise} \end{array} \right. \ .$$ (27)

Whenever $\phi < 1$ the local reconstruction may be modified according to

$$\widetilde u_i = \overline u_i + \mathcal I^1(\wt{ \Delta u})(x) \ ,$$ (28)

where $\mathcal I^1$ is the linear interpolation operator defined by the nodes $x_r$, and

$$\wt{ \Delta u(x_r)} = \phi(x_r)\left( \mathcal P(u_i)(x_r) - \overline u_i \right) \ ,$$ (29)

A modification may be introduced by rejecting any limiting whenever $\Delta u < M (\Delta x)^2$, where $\Delta x$ is a characteristic mesh length. This will prevent a modification of the solution near smooth extrema and is related to the theory of total-variation boundedness (TVB) [3]. In practice we usually choose values of $M$ between 0 and 40.

The procedure is sufficient to ensure that the reconstruction stays within the prescribed bounds, but it is not conservative in the sense that in general

$$\int_{T_i} \widetilde u_i \, dx \neq \int_{T_i} u_i \, dx \ .$$ (30)

A further modification of the linear solution is proposed in [5] , which ensures conservation in a way that prevents an increase of the slope. We adopt this strategy for the present work, omitting the technical details for the sake of brevity, and refer instead to [5]. Our general procedure becomes identical to the one described in [5] if a cell-wise neighborhood is used to compute the maximum allowed slope. In the present work we use the nodal neighborhood, and the full reconstruction is evaluated at the nodes of the triangles and compared to the reference state.