Correct Formula for Potential Energy

[DanielDoehring]

The formula for the (gravitational) potential energy from the Euler-Gravity paper is wrong:

and should contain a factor of 0.5 according to the book "Galactic Dynamics"

The reason why this was not spotted is that the analytical solution in the reproducibility repository is only read from file, not computed from the formula for the analytical solution (see e.g. FLASH user guide, equation (34.35))

$$E_\text{pot} = -\frac{\pi G (\rho_0 \delta_0 L)^2}{2 k^2} (1 + \cos(2 \omega t) )$$

Using this code

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import ScalarFormatter

df_Jeans = pd.read_csv('./analysis.dat', delim_whitespace=True, skiprows=1, header=None)

RWTH_Blue_RGB   = [(0, 84/256, 159/256)]

fig, ax = plt.subplots()

# .dat File Structure:
#timestep time dt e_total e_kinetic e_internal e_potential 

# Parameters as found in elixir
rho0 = 1.5e7
p0 = 1.5e7
gamma = 5/3
c0 = np.sqrt(gamma * p0 / rho0)
G = 6.674e-8
delta0 = 1e-3
L = 1.0 # Domain length

k = 4 * np.pi

omega_squared = (c0 * k)**2 - 4 * np.pi * G * rho0
omega = np.sqrt(omega_squared)

# Skip first entry (wrong result for naive initialization which does not fulfill the Poisson equation)
time = df_Jeans[1][1:-1]

E_potential_analytical = - np.pi * G / 2 * (rho0 * delta0 * L / k)**2 * (1 + np.cos(2 * omega * time))

#plt.plot(time, df_Jeans[6][1:-1], color = RWTH_Blue_RGB[0], label = "Numerical Approximation")
plt.scatter(time, df_Jeans[6][1:-1], color = RWTH_Blue_RGB[0], label = "Numerical Approximation")

plt.plot(time, E_potential_analytical, color = 'k', label = "Analytical Solution")

plt.title("Potential Energy")

formatter = ScalarFormatter(useMathText=True)
formatter.set_scientific(True)
ax.xaxis.set_major_formatter(formatter)
ax.yaxis.set_major_formatter(formatter)

plt.legend(loc = 'lower left')

ax.grid(axis ='y', which='both', alpha=0.1, linewidth = 1.5, color ='black')
ax.grid(axis ='x', which='both', alpha=0.1, linewidth = 1.5, color ='black')
ax.set_axisbelow(True)  # Hide grid 


### Crop plot window ###
ax.set_xlim([3, 5])

ax.set_xlabel(r'$t$')

plt.tight_layout()  # Title, labels, ... to screen

plt.show()

one obtains the correct plot:

[github-actions]

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[andrewwinters5000]

Thanks for catching this!

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 83.95%. Comparing base (8592c0e) to head (a52a659).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #2227      +/-   ##
==========================================
+ Coverage   77.62%   83.95%   +6.34%     
==========================================
  Files         486      486              
  Lines       39107    39118      +11     
==========================================
+ Hits        30353    32840    +2487     
+ Misses       8754     6278    -2476     

Flag	Coverage Δ
unittests	83.95% <100.00%> (+6.34%)	⬆️

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