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\section{Limiting}



Let $w, z, h$  be the stage, bed elevation and depth at the centroid and
let $w_i, z_i, h_i$ be the stage, bed elevation and depth at vertex $i$.

Define the maximal bed elevation range $dz$ as

\[
  dz = \max_i |z_i - z|
\]

and the minimal depth $h_{\mbox{\tiny min}}$ as

\[
  h_{\mbox{\tiny min}} = \min_i h_i
\]


\[
  \alpha = \left \{
  \begin{array}{ll} 
    \max (\min ( 2 h_{\mbox{\tiny min}} / dz )) & dz > 0 \\
    1 & dz \leq 0     
  \end{array}
  \right .  
\]


Let $\tilde{w_i}$ be the stage obtained from a gradient limiter limiting on stage. The corresponding depth is the defined as

\[
  \tilde{h_i} = \tilde{w_i} - z_i
\]

Let $\bar{h_i}$ be the depth obtained from a gradient limiter limiting on depth.
The corresponding stage is the defined as

\[
  \bar{w_i} = z_i + \bar{h_i}
\]


The balanced stage $w_i$ is then obtained by the linear combination

\[
  w_i = \alpha \tilde{w_i} + (1-\alpha) \bar{w_i}
\]

or

\[
  w_i = z_i + \alpha \tilde{h_i} + (1-\alpha) \bar{h_i}
\]

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