Back in the summer of 2021, severe floods struck Western Germany, resulting in several cities and villages being destroyed and hundreds of people losing their lives. Modelling and predicting those floods can be crucial for investigating such natural disasters. The shallow water equations are a system of physical conservation equations that can model water flow over a given domain. The two dimensional shallow water equations are given as follows:<\/p>\n
\\begin{equation*}
\n \\left\\{\\begin{aligned}
\n \\frac{\\partial}{\\partial t} h + \\left( h v_1\\right)_x + \\left( h v_2\\right)_y =0&\\\\
\n \\frac{\\partial}{\\partial t} hv_1 + \\left( hv_1^2 + \\frac{1}{2} gh^2\\right)_x + \\left(h v_1v_2\\right)_y &= -ghb_x\\\\
\n \\frac{\\partial}{\\partial t}h v_2 + \\left( hv_1v_2\\right)_x + \\left(hv_2^2 +\\frac{1}{2} gh^2\\right)_y &= -ghb_y
\n \\end{aligned}\\right.
\n\\end{equation*}<\/p>\n
Here the variable $b$ describes the underlying bottom topography function of the domain. As this system is set up of hyperbolic partial differential equations, we use the numerical solver Trixi.jl<\/a> to approximate the solutions. This solver is part of the trixi-framework<\/a>. Trixi.jl already implemented the shallow water equations but did not yet offer the functionality to include real-world topography data into the calculations. Therefore the add-on TrixiBottomTopography.jl<\/a> has been included in the trixi-framework. This tool uses B-spline interpolation approaches to remodel the bottom topography function from real-world data, which is often given as elevation data at specific points. One example is the DGM1<\/a> data set which provides uniformly spaced elevation data of the whole German state of North Rhine-Westphalia. Using this data set and the functionalities implemented in Trixi.jl as well as TrixiBottomTopography.jl, we can simulate a dam break problem over a section of the Rhine river valley.<\/p>\n Back in the summer of 2021, severe floods struck Western Germany, resulting in several cities and villages being destroyed and hundreds of people losing their lives. Modelling and predicting those floods can be crucial for investigating such natural disasters. The … Continue reading ![\"\"](\"https:\/\/www.mi.uni-koeln.de\/NumSim\/wp-content\/uploads\/2022\/12\/Dam_break-1024x552.gif\")
<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"