This example illustrates solving differential equations numerically in Julia. Specifically, it solves an ODE after an example taken from this Python notebook, using DifferentialEquations.jl
ode_test.jl: Julia source codefigure.py: Python script for generating the figurerun.sbatch: Batch-job submission scriptfigure.png: Figure of ODE's solution
#++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# Program: ode_test.jl
#++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
using DifferentialEquations
using SimpleDiffEq
using Printf
using Plots
using Plots.PlotMeasures
using PyCall
# --- Define the problem ---
f(u, p, t) = t - u
u0 = 1.0 # Initial conditions
tspan = (0.0, 5.0) # Time span
prob = ODEProblem(f, u0, tspan)
# --- Solve the problem ---
dt = 0.2
alg = RK4()
sol = solve(prob, alg, saveat=dt)
# --- Exact solution ---
function f_ex(t)
f = t - 1.0 + ( 2.0 * exp(-t) )
return f
end
# --- Print out solution ---
fname = "results.dat"
fo = open(fname, "w")
println(fo, " Time Numeric Exact ")
println(fo, " ------------------------------ ")
for i = 1: size(sol.t)[1]
x = sol.t[i]
y = sol.u[i]
y_exact = f_ex(x)
@printf(fo, "%8.4f %10.6f %10.6f\n", x, y, y_exact)
end
close(fo)
pyimport("numpy")
# --- Run the Python script to plot the result ---
@pyinclude("figure.py")Note: if not already installed, you will need to install the
DifferentialEquations,SimpleDiffEq,Plots,PyCall,Conda, andmatplotlibpackages in order to runode_test.jlsuccessfully. Follow these instructions to install the packages.
#!/bin/bash
#SBATCH -J ode_test
#SBATCH -o ode_test.out
#SBATCH -e ode_test.err
#SBATCH -p test
#SBATCH -N 1
#SBATCH -c 1
#SBATCH -t 0-00:30
#SBATCH --mem=4G
# Run the program using Julia
julia ode_test.jlsbatch run.sbatch"""
Program: figure.py
"""
import os
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator, FormatStrFormatter
def rc_params():
"""Set rcParams for this plot"""
params = {
'axes.linewidth':2.0,
'xtick.major.size':6.5,
'xtick.major.width':1.5,
'xtick.minor.size':3.5,
'xtick.minor.width':1.5,
'ytick.major.size':6.5,
'ytick.major.width':1.5,
'ytick.minor.size':3.5,
'ytick.minor.width':1.5,
'xtick.labelsize':22,
'ytick.labelsize':22,
'xtick.direction':'in',
'ytick.direction':'in',
'xtick.top':True,
'ytick.right':True
}
mpl.rcParams.update(params)
# Set rcParams
rc_params()
# Data
cwd = os.getcwd()
data_path = os.path.join(cwd, 'results.dat')
# Figure
fig_path = os.path.join(cwd, 'figure.png')
# Load data
darr = np.loadtxt(data_path, skiprows=2)
t = darr[:,0]
y = darr[:,1]
y_ex = darr[:,2]
# Plot results
fig, ax = plt.subplots(figsize=(8,6))
p1, = ax.plot(t, y_ex, linewidth = 3.0, color="red", alpha=0.5,
linestyle='--', label='Exact Solution')
p2, = ax.plot(t, y_ex, linewidth = 3.0, color="blue", alpha=0.5,
marker='o', markersize=7, linestyle='', label='Numeric Solution')
plt.xlim([0.0, 5.0])
plt.ylim([0.0, 4.3])
plt.xlabel('Time', fontsize=22)
plt.ylabel('u(t)', fontsize=22)
plt.legend(fontsize=15, loc="upper left", shadow=True, fancybox=True)
plt.savefig(fig_path, format='png', dpi=100, bbox_inches='tight')$ cat results.dat
Time Numeric Exact
------------------------------
0.0000 1.000000 1.000000
0.2000 0.837462 0.837462
0.4000 0.740639 0.740640
0.6000 0.697624 0.697623
0.8000 0.698660 0.698658
1.0000 0.735763 0.735759
1.2000 0.802393 0.802388
1.4000 0.893198 0.893194
1.6000 1.003802 1.003793
1.8000 1.130613 1.130598
2.0000 1.270681 1.270671
2.2000 1.421625 1.421606
2.4000 1.581455 1.581436
2.6000 1.748569 1.748547
2.8000 1.921645 1.921620
3.0000 2.099603 2.099574
3.2000 2.281551 2.281524
3.4000 2.466785 2.466747
3.6000 2.654677 2.654647
3.8000 2.844781 2.844742
4.0000 3.036672 3.036631
4.2000 3.230027 3.229991
4.4000 3.424601 3.424555
4.6000 3.620150 3.620104
4.8000 3.816498 3.816459
5.0000 4.013510 4.013476