sth wrong with the numerical solution

moving_domain/moving_time_dependent_alpha.py

@@ -29,7 +29,7 @@ def advection_diffusion(Nx, L, Nt, tmax, D):
     c_array[0] = c0.compute_vertex_values(mesh)
 
     # velocity
-    alpha = df.Expression('times < 5 ? 0.01 : 0.0', times = 0, degree=0)
+    alpha = df.Expression('times < 1 ? alpha0 : 0', times = 0, alpha0 = alpha0, degree=0)
 
     v = df.Expression('alpha*x[0]', alpha=alpha, degree=1)
     u = df.interpolate(v, SFS)
@@ -41,24 +41,24 @@ def advection_diffusion(Nx, L, Nt, tmax, D):
 
     # 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]
+        alpha.times = times[i]
 
     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
+dx, L, dt, tmax, D = 0.01, 1, 0.01, 10, 0.1
+alpha0, tc = 1, 1
 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]
 
@@ -66,11 +66,13 @@ c = advection_diffusion(Nx, L, Nt, tmax, D)[0]
 c_exact = np.zeros((Nt+1, Nx+1))
 for j in range(0, len(times)):
     if times[j] <= tc:
+        # diffusion on moving domain with velocity alpha0*x
         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: 
+        # diffusion on fixed domain of length L = L0*exp(alpha0 * tc) with initial condition to be the profile at tc
         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))
@@ -80,17 +82,17 @@ 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)
+ax.set_ylim(min(np.min(c), np.min(c_exact)), max(np.max(c), np.max(c_exact)))
 
 cplot, = ax.plot(x[0], c[0], '--',label = 'Numerical solution')
-c_exactplot, = ax.plot(x[0], c_exact[0],label = 'Exact 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])
+        # 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])