Task 3 Consider the one-dimensional advection-diffusion equation: ac ac at to be solved with the boundary and initial conditions: C(0, t) = 0 C(L, t) = 100 C(x, 0) = 100r L OSIEL. The analytical solution is: C(x. t) = 1001 Anett sinh () EP - 1 where P is the Peclet number P = T and the coefficients Ax, Bx are given by: Ak = (-1)*_- sin e-Akt BK T BK = (-1)#+12 (1+ 3) er + sin where BK = 2 ) + ( RX ) ? DBk XK = L2 Here, C(r, t) is the concentration of the pollutant at point r and time t, u is the constant wind speed in the r direction and D is the diffusivity coefficient in the r direction. Assume that L = 1.0, D = 0.01, u =0.1, Ar = 0.1(i.e. N =10) and At = 0.01. 3.1 Discretize the domain of C to a grid. 3.2 Use the Forward Time Centred Space scheme to discretize the given partial differential equation. 2 3.3 Discretize the boundary conditions. 3 3.4 Evaluate the consistency of the Forward Time Centred Space scheme when applied to the given partial differential equation. 5 3.5 Compare the numerical solution with exact solution at time t = 0, t = 2, t = 5 and t = 10. Plot on the same figure. 6 3.6 Plot the absolute error, which is the absolute difference between exact and numerical solution, at time t =0, t = 2, t =5 and t = 10. Plot on the same figure. 6