\documentclass[12pt]{article} \usepackage{graphicx} \begin{document} \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 \[ w_i = z_i + \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} \] \end{document}