Task 3 Consider the one-dimensional advection-diffusion equation: ac ac a-c at to be solved with the boundary and initial conditions: C(0, t) = 0 C(L, t) = 100 C(x, 0) = - 100x OSISL. The analytical solution is: C(x, t) = 100 et - 1 4near sinh (7) PP - 1 er - 1 LAK + 2Test ) BR K = 1 where P is the Peclet number UL P = D and the coefficients Ap, Bk are given by: Ak = (-1)k k sin kux e-dat BK BK = (-1)*+12 ( 1 +2)ekp] 32 sin KNx e- Axt where BK = + ( KI ) ? DBk L2 Here, C(x, t) is the concentration of the pollutant at point a and time t, u is the constant wind speed in the x direction and D is the diffusivity coefficient in the r direction. Assume that L = 1.0, D = 0.01, u = 0.1, Ax = 0.1(i.e. N = 10) and At = 0.01. 3.1 Discretize the domain of C to a grid. 3.2 Use the Crank-Nicolson method to discretize the given partial differential equation. 12 3.3 Discretize the initial and boundary conditions. 3 3.4 Compare the numerical solution with exact solution at time t = 0, t = 1, t = 2 and t = 5. Plot on the same figure. 6 3.5 Plot the absolute error at time t = 5 when Ax = 0.1, Ar = 0.05 and Ax = 0.025. Plot on the same figure. 6