Shallow Water Equations Solver

This package provides a well-balanced solver for the one-dimensional Saint-Venant equations, based on the principles outlined in this paper and this presentation.

Installation

The package is available through pip, and may be installed via:

pip install SWE_Solver

Main Usage

To utilize this package, you can call the plotSWE function with the following parameters:

h, u = plotSWE(B, h0, u0, Nx, tEnd, timePoints, g=1, method='C')

Parameters:

• B (callable): Bottom topography function. This function defines the topographic profile and should take spatial coordinates as input and return the bottom elevation at those coordinates.
• h0 (array): Initial water height profile. This should be an array of length Nx, representing the initial water height at different spatial locations.
• u0 (array): Initial water velocity profile. Similar to h0, this should be an array of length Nx, representing the initial water velocity at different spatial locations.
• Nx (int): Number of spatial grid points.
• tEnd (float): End time of the simulation. The simulation starts at time t=0.
• timePoints (list): List of time points at which you want to visualize the results.
• g (float, optional): Gravitational constant. Default is 1.
• method (str, optional): Method selection ('A', 'B' or 'C'). Default is 'C'.

Returns:

• h (array): Array containing the water height profile at the final time point.
• u (array): Array containing the water velocity profile at the final time point.

Pre-Coded Examples

A number of pre-coded examples are available through the library, through the function exampleSWE.

h, u = exampleSWE(state="still_flat", method='C')

Parameters

• state (String): Name of the example. Has to be one of "still_flat" (Constant height, zero velocity, flat bottom), "still_tilted" (Constant total height, zero velocity, tilted bottom),, "still_tilted_pert" (Perturbed constant total height, perturbed zero velocity, tilted bottom), "moving_flat" (Constant height, constant velocity, flat bottom), "moving_tilted" (Constant total height, constant velocity, tilted bottom), "evolving_wave" (Step function for height, constant discharge, flat bottom), "standing_wave" (Final profile of "evolving_wave" for method 'C', representing an equilibrium), "standing_wave_pert" (Final profile of "evolving_wave" for method 'C', with a perturbation), "forming_collision" (Constant water height, positive velocity on the right, negative velocity on the left, flat bottom), "spike_flattening" (Water height given by a Gaussian, zero velocity, flat bottom), "over_bump" (Constant total water height, constant velocity, bottom given by a Gaussian). Defaults to "still_flat".
• method (String): Name of the method used. Has to be one of 'A', 'B', 'C'. Defaults to 'C'.

Returns

• h (array): Array containing the water height profile at the final time point.
• u (array): Array containing the water velocity profile at the final time point.

Example

from SWE_Solver import plotSWE
from math import sqrt
from scipy.special import erf

Nx = 50
B = lambda x: 1
f = lambda T: 1 + sqrt(3) / (1 - erf(-0.5 / 0.1)) * (erf((T - 0.5) / 0.1) - erf (-0.5 / 0.1))
h0 = [f(_/ (Nx-1)) for _ in range(Nx)]
u0 = [2.0 / h0[_] for _ in range(Nx)]
_ = plotSWE(B, h0, u0, Nx, tEnd=1.0, timePoints=[0.0, 0.1, 0.5, 1.0])

The above is equivalent to the simple example given by

from SWE_Solver import exampleSWE

_ = exampleSWE("evolving_wave", 'C')

In this example, we're using a spatial grid with 50 points, running the simulation up to t=1 seconds, and visualizing the results at times 0.0, 0.1, 0.5 and 1.0 seconds, with gravitational constant g=1 (default value) and using method='C' (default value).

This produces the result in the following figure.