[SOLVED] AA543 - Homework 2

Description

Given the 1D heat conduction problem:

u_t = αu_xx                                                                                                                                      (1)

where u = u(x,t) and α = 1, with initial conditions (ICs):

if      0 ≤ x ≤ 0.5

(2)

if       0.5 < x ≤ 1.0

and boundary conditions (BCs):

u(x = 0,t) = u(x = 1,t) = 0                                                                        (3)

in the interval 0 ≤ x ≤ 1, do the following:

1.1 Derive the analytical solution of the problem stated above.

1.2 Write a computer program (and on paper the numerical algorithm) to find the numerical solution of the 1D heat conduction problem using the following schemes:

1. one-step forward in time and 2nd-order central difference in space (explicit);
2. one-step backward in time and 2nd-order central difference in space (implicit);
   • Compute the numerical solution for ∆x = 0.05, and for ∆t = 0.0012 and ∆t = 0.0013, and plot the numerical solution along with the analytical solution at t = 0, ∆t, 10∆t, 50∆t.
   • Discuss your solutions, and if your code does not work explain why.
   • Advanced (Bonus Points): Extend the 1D heat conduction problem to 2D, formulate the problem in 2D and solve 1.1 to 1.4 above in the 2D case.

1

Problem 2:

Given the 1D linear convection problem:

u_t + au_x = 0                                                                                      (4)

where u = u(x,t) and a = 0.5, with initial conditions (ICs):

1         if −7 ≤ x ≤−5

(         = 0) =                                                                                                                  (5)

0    if        x < −7,x > −5

and boundary conditions (BCs):

u(x = −10,t) = 0                                                                                 (6)

in the interval −10 ≤ x ≤ 10, do the following:

2.1 Write a computer program (and on paper the numerical algorithm) to find the numerical solution of the 1D linear convection problem using the following schemes: 1. one-step forward in time (explicit) and first-order upwind in space;

2. Lax-Wendroff scheme.
3. an implicit scheme of your choice.
   • Compute the numerical solution for different CFL numbers, CFL = 0.6 and 1.2, and plot the numerical solutions along with the analytical solution [ u(x,t) = u(x − at) ] at t = t_max/2 where t_max is the time at which the square wave reaches the right boundary x = 10.
   • Discuss your solutions, and if your code does not work explain why.
   • Derive and plot the dispersion and diffusion errors of the three numerical schemes above at two different CFL numbers of your choice.
   • Advanced (Bonus Points): Extend the 1D linear convection problem to 2D, formulate the problem in 2D, and solve 2.1 to 2.3 above in the 2D case