"""
This code solves the advection equationU_t + vU_x = 0over the spatial domain of 0 <= x <= 1 that is discretized
into 103 nodes, with dx=0.01, using the First-Order Upwind (FOU)
scheme with Forward difference in Eq.
for an initial profile of a Gaussian curve, defined by U(x,t) = exp(-200*(x-xc-v*t).^2)where xc=0.25 is the center of the curve at t=0.The periodic boundary conditions are applied either end of the domain.
The velocity is v=1. The solution is iterated until t=1.5 seconds.
"""import numpy as np
import matplotlib.pyplot as plt
import timestart =time.clock()class UpwindMethod1:def __init__(self, N, tmax):self.N = N # number of nodesself.tmax = tmaxself.xmin = 0self.xmax = 1self.dt = 0.009 # timestepself.v = 1 # velocityself.xc = 0.25self.initializeDomain()self.initializeU()self.initializeParams()def initializeDomain(self):self.dx = (self.xmax - self.xmin)/self.Nself.x = np.arange(self.xmin-self.dx, self.xmax+(2*self.dx), self.dx)def initializeU(self):u0 = np.exp(-200*(self.x-self.xc)**2)self.u = u0.copy()self.unp1 = u0.copy()def initializeParams(self):self.nsteps = round(self.tmax/self.dt)def solve_and_plot(self):tc = 0for i in range(self.nsteps):plt.clf()# The FOU scheme, Eq. (18.21)for j in range(self.N+2):self.unp1[j] = self.u[j] - (self.v*self.dt/(self.dx))*(self.u[j]-self.u[j-1]) self.u = self.unp1.copy()# Periodic boundary conditionsself.u[0] = self.u[self.N+1]self.u[self.N+2] = self.u[1]plt.plot(self.x, self.u, 'bo-', label="First-order Upwind")plt.axis((self.xmin-0.12, self.xmax+0.12, -0.2, 1.4))plt.grid(True)plt.xlabel("Distance (x)")plt.ylabel("u")plt.legend(loc=1, fontsize=12)plt.suptitle("Time = %1.3f" % (tc+self.dt))plt.pause(0.01)tc += self.dtdef main():sim = UpwindMethod1(100, 1.5)sim.solve_and_plot()plt.show()if __name__ == "__main__":main()end = time.clock()print('Running time: %s Seconds'%(end-start))