alpha=1-theta function

moving_domain/moving_time_dependent_alpha.py

+import dolfin as df
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.widgets import Slider
+import progressbar
+
+df.set_log_level(df.LogLevel.ERROR)
+df.parameters['form_compiler']['optimize'] = True
+
+def advection_diffusion(Nx, L, Nt, tmax, D):
+    # mesh, function space, function, test function
+    mesh = df.IntervalMesh(Nx, 0, L)
+    SFS = df.FunctionSpace(mesh, 'P', 1)
+    c = df.Function(SFS)
+    tc = df.TestFunction(SFS)
+
+    # x and t arrays
+    times = np.linspace(0, tmax, Nt+1)
+    dt = times[1] - times[0]
+
+    # initial condition
+    c0 = df.Function(SFS)
+    c0.interpolate(df.Expression('1 + 0.2 * cos(pi*x[0]/L)', pi=np.pi, L=L, degree=1))
+
+    # arrays
+    c_array = np.zeros((Nt+1, Nx+1))
+    x_array = np.zeros((Nt+1, Nx+1))
+    x_array[0] = mesh.coordinates()[:, 0]
+    c_array[0] = c0.compute_vertex_values(mesh)
+
+    # velocity
+    alpha = df.Expression('times < 5 ? 0.01 : 0.0', times = 0, degree=0)
+
+    v = df.Expression('alpha*x[0]', alpha=alpha, degree=1)
+    u = df.interpolate(v, SFS)
+
+    # form
+    cform = (df.inner((c - c0)/dt, tc) 
+            + D * df.inner(df.nabla_grad(c), df.nabla_grad(tc))
+            + df.inner((u*c).dx(0), tc) )* df.dx
+
+    # solve
+    for i in progressbar.progressbar(range(1, Nt+1)):
+        alpha.times = times[i]
+        df.solve(cform == 0, c)
+        c_array[i] = c.compute_vertex_values(mesh)
+        c0.assign(c)
+       
+        df.ALE.move(mesh, df.Expression('v*dt', v=v, dt=dt, degree=1))
+        x_array[i] = mesh.coordinates()[:,0]
+
+    return c_array, x_array
+
+# plot c(x,t) numerical and analytical for given dt
+dx, L, dt, tmax, D = 0.001, 1, 0.001, 20, 0.01
+alpha0, tc = 0.01, 5
+Nx = int(L/dx)
+Nt = int(tmax/dt)
+
+times = np.linspace(0, tmax, Nt+1)
+x = advection_diffusion(Nx, L, Nt, tmax, D)[1]
+# numerical solution
+c = advection_diffusion(Nx, L, Nt, tmax, D)[0]
+
+# analytical solution
+c_exact = np.zeros((Nt+1, Nx+1))
+for j in range(0, len(times)):
+    if times[j] <= tc:
+        alpha = alpha0
+        l = L * np.exp(alpha * times[j])
+        beta = (-D * np.pi**2 /(2 *alpha * L**2)) * (1 - np.exp(-2 * alpha * times[j]))
+        c_exact[j] = np.exp(-alpha * times[j])* (1 + 0.2 * np.cos(np.pi*x[j]/l) * np.exp(beta))
+    else:
+        l = L * np.exp(alpha0 * tc)
+        beta = -D * np.pi**2*times[j]/l**2 - np.pi**2 * D/(2*l**2*alpha0) * (1 - np.exp(-2 *alpha0 *tc))
+        c_exact[j] = np.exp(-alpha0 * tc) * (1 + 0.2 * np.exp(beta) * np.cos(np.pi * x[j]/l))
+
+
+fig, ax = plt.subplots(1,1,figsize=(8,6))
+ax.set_xlabel(r'$x$')
+ax.set_ylabel(r'$c(x,t)$')
+ax.set_xlim(np.min(x), np.max(x)+2)
+ax.set_ylim(np.min(c)-1, np.max(c)+1)
+
+cplot, = ax.plot(x[0], c[0], '--',label = 'Numerical solution')
+c_exactplot, = ax.plot(x[0], c_exact[0],label = 'Exact solution')
+
+def update(value):
+        ti = np.abs(times-value).argmin()
+        cplot.set_xdata(x[ti])
+        cplot.set_ydata(c[ti])
+        c_exactplot.set_xdata(x[ti])
+        c_exactplot.set_ydata(c_exact[ti])
+        plt.draw()
+        
+sax = plt.axes([0.1, 0.92, 0.7, 0.02])
+slider = Slider(sax, r'$t/\tau$', min(times), max(times),
+        valinit=min(times), valfmt='%3.1f',
+        fc='#999999')
+slider.drawon = False
+slider.on_changed(update)
+ax.legend(loc=0)
+plt.show()
+