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[1763] | 1 | \documentclass[12pt]{article} |
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| 2 | \usepackage{graphicx} |
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| 3 | |
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| 4 | |
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| 5 | \begin{document} |
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| 6 | |
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| 7 | |
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| 8 | \section{Limiting} |
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| 9 | |
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| 10 | |
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| 11 | |
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| 12 | Let $w, z, h$ be the stage, bed elevation and depth at the centroid and |
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| 13 | let $w_i, z_i, h_i$ be the stage, bed elevation and depth at vertex $i$. |
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| 14 | |
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| 15 | Define the maximal bed elevation range $dz$ as |
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| 16 | |
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| 17 | \[ |
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| 18 | dz = \max_i |z_i - z| |
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| 19 | \] |
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| 20 | |
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[1778] | 21 | and the minimal depth $h_{\mbox{\tiny min}}$ as |
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[1763] | 22 | |
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| 23 | \[ |
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[1778] | 24 | h_{\mbox{\tiny min}} = \min_i h_i |
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[1763] | 25 | \] |
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| 26 | |
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| 27 | |
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| 28 | \[ |
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| 29 | \alpha = \left \{ |
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| 30 | \begin{array}{ll} |
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[1778] | 31 | \max (\min ( 2 h_{\mbox{\tiny min}} / dz )) & dz > 0 \\ |
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[1763] | 32 | 1 & dz \leq 0 |
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| 33 | \end{array} |
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| 34 | \right . |
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| 35 | \] |
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| 36 | |
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| 37 | |
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| 38 | Let $\tilde{w_i}$ be the stage obtained from a gradient limiter limiting on stage. The corresponding depth is the defined as |
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| 39 | |
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| 40 | \[ |
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| 41 | \tilde{h_i} = \tilde{w_i} - z_i |
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| 42 | \] |
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| 43 | |
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| 44 | Let $\bar{h_i}$ be the depth obtained from a gradient limiter limiting on depth. |
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| 45 | The corresponding stage is the defined as |
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| 46 | |
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| 47 | \[ |
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| 48 | \bar{w_i} = z_i + \bar{h_i} |
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| 49 | \] |
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| 50 | |
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| 51 | |
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| 52 | The balanced stage $w_i$ is then obtained by the linear combination |
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| 53 | |
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| 54 | \[ |
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| 55 | w_i = \alpha \tilde{w_i} + (1-\alpha) \bar{w_i} |
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| 56 | \] |
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| 57 | |
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| 58 | or |
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| 59 | |
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| 60 | \[ |
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| 61 | w_i = z_i + \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} |
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| 62 | \] |
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| 63 | |
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| 64 | \end{document} |
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