Elevation model (Water Overlay): Difference between revisions
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The first requirement the scheme to become ''well balanced and positivity preserving'' is to ensure that each grid cell has a constant linear slope in both the x- and y- direction. Secondly the end points of the slope should meet in the center of the cell's edges. This ensures that the bottom is continuous along cells in the x- and y- direction. Thirdly, the linear slope in the x- and y-direction within a cell should meet in a single center point. | The first requirement the scheme to become ''well balanced and positivity preserving'' is to ensure that each grid cell has a constant linear slope in both the x- and y- direction. Secondly the end points of the slope should meet in the center of the cell's edges. This ensures that the bottom is continuous along cells in the x- and y- direction. Thirdly, the linear slope in the x- and y-direction within a cell should meet in a single center point. | ||
To fulfill these requirements, the following steps are | To fulfill these requirements, the following steps are taken: | ||
# Pick or calculate the height points for the 4 corners of the cell. | # Pick or calculate the height points for the 4 corners of the cell. | ||
# Form a rectangle with the 4 corners and calculate the centers of these edges. (These are the points that have to meet for continuity) | # Form a rectangle with the 4 corners and calculate the centers of these edges. (These are the points that have to meet for continuity) | ||
# Calculate a new center point based on the 4 edge center points. | # Calculate a new center point based on the 4 edge center points. | ||
Given that the adjacent cells share the same corner points, and thus share an edge center point, the bottom will be continuous in the x and y direction. | Given that the adjacent cells share the same corner points, and thus share an edge center point, the bottom will be continuous in the x and y direction. Furthermore, the cell has an linear slope in both the x- and y-direction. The only downside is that the new center point might have been placed heigher or lower in a situation where the terrain's slope was originally not linear within the cell. | ||
Furthermore, the cell has an linear slope in both the x- and y-direction. | |||
The only | |||
<ul> | <ul> | ||
<li style="display: inline-block;">[[File:Inundation overlay 04 HWP(2).PNG|left|thumb|400px|2D linear piecewise reconstruction. Source: Horváth et al. (2014)<ref name="Horvath"/>]]</li> | <li style="display: inline-block;">[[File:Inundation overlay 04 HWP(2).PNG|left|thumb|400px|2D linear piecewise reconstruction. Source: Horváth et al. (2014)<ref name="Horvath"/>]]</li> | ||
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===Notes=== | ===Notes=== | ||
* The smaller the [[grid cell size|grid size]], the closer the bottom reconstruction will approximate the original surface elevation. | * The smaller the [[grid cell size|grid size]], the closer the bottom reconstruction will approximate the original [[terrain height (Water Overlay)|surface elevation]]. | ||
* The resulting elevation model can be inspected in a project using the [[Surface elevation result type (Water Overlay)|SURFACE_ELEVATION]] result type. | |||
==References== | ==References== | ||
Revision as of 13:50, 18 June 2019
The elevation model is :
- Rasterization of the height sectors
- Piecewise linear reconstruction of the bottom
Piecewise linear reconstruction of the bottom
The implementation of the Water Module is based on second-order semi-discrete central-upwind scheme by Kurganov and Petrova (2007)[1]. The surface elevation, also named bottom in the paper, is slightly adjusted to support the scheme to become well balanced and positivity preserving. The process of adjusting the original surface elevation is called piecewise linear reconstruction of the bottom.
The first requirement the scheme to become well balanced and positivity preserving is to ensure that each grid cell has a constant linear slope in both the x- and y- direction. Secondly the end points of the slope should meet in the center of the cell's edges. This ensures that the bottom is continuous along cells in the x- and y- direction. Thirdly, the linear slope in the x- and y-direction within a cell should meet in a single center point.
To fulfill these requirements, the following steps are taken:
- Pick or calculate the height points for the 4 corners of the cell.
- Form a rectangle with the 4 corners and calculate the centers of these edges. (These are the points that have to meet for continuity)
- Calculate a new center point based on the 4 edge center points.
Given that the adjacent cells share the same corner points, and thus share an edge center point, the bottom will be continuous in the x and y direction. Furthermore, the cell has an linear slope in both the x- and y-direction. The only downside is that the new center point might have been placed heigher or lower in a situation where the terrain's slope was originally not linear within the cell.
2D linear piecewise reconstruction. Source: Horváth et al. (2014)[2]
Application of reconstructed elevation Source: Horváth et al. (2014)[2]
.PNG)
.PNG)
Notes
- The smaller the grid size, the closer the bottom reconstruction will approximate the original surface elevation.
- The resulting elevation model can be inspected in a project using the SURFACE_ELEVATION result type.
References
- ↑ 1.0 1.1 Kurganov A, Petrova G (2007) ∙ A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System ∙ found at: http://www.math.tamu.edu/~gpetrova/KPSV.pdf (last visited 2018-06-29)
- ↑ 2.0 2.1 Zsolt Horváth, Jürgen Waser, Rui A. P. Perdigão, Artem Konev and Günter Blöschl (2014) ∙ A two-dimensional numerical scheme of dry/wet fronts for the Saint-Venant system of shallow water equations ∙ found at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.700.7977&rep=rep1&type=pdf ∙ http://visdom.at/media/pdf/publications/Poster.pdf ∙ (last visited 2018-06-29)
