Discretize the stationary heat equation

Fakultät für Mathematik und Informatik 26. Juni 2013 TU Bergakademie Freiberg

Prof. Dr. O. Rheinbach/Dr. M. Helm

Numerical Analysis of Differential Equations Boundary Value Problems (II)

Exercise 1

Discretize the stationary heat equation

−∆u(x, y) = −3 ((x, y) ∈ Ω) on the following mesh:

1 4 7

2 3

5 6

8 9

10 11 12 13

14 15 16 17

18 19

a) Use u = 0 as Dirichlet boundary values on the open circles and u = 1 on the open squares.

b) Replace the Dirichlet boudary condition on the left and on the bottom-right by

= 0 on the lines 1 3 and 17 19.

Exercise 2

A rectangular silver plate of 6 cm×5 cm is heated uniformly at each point with a constant energy rate q = 6.2802 W/cm

. Assume that the temperature along the boundaries is hold at

u(x, 0) = x(6 − x), u(x, 5) = 0 (0 < x < 6), u(0, y) = y(5 − y), u(6, y) = 0 (0 < y < 5).

Then, the stationary temperature distribution can be described by the Poisson equation

∆u(x, y) = − q

K for (x, y) ∈ Ω := (0, 6) × (0, 5),

together with the above boundary conditions. The constant K = 4.3543 W/(cm·K) is the thermal conductivity.

Approximate the temperature distribution by using the finite difference method with mesh widths

∆x = 0.4 and ∆y =

. Try also some other mesh widths.

Initial Value Problems II 2

Exercise 3

Consider the initial-boundary value problem

u

(x, t) −

u

(x, t) = 0 in Ω = (0, 1) × (0, T ], u(x, 0) = x(1 − x) für x ∈ [0, 1],

u(0, t) = u(1, t) = 0 für t ∈ [0, T ] for an unknown function u = u(x, t).

a) Approximate u = u(x, t) at time t = 0.1 with an explicit difference scheme. Take a step size of ∆x = 0.25 in space and ∆t = 0.1 in time.

b) Approximate u = u(x, t) at time t = 0.1 with a purely implicit difference scheme. Take the same step sizes as in (a).

c) For which step sizes ∆t in time is the explicit and the purely implicit difference scheme

numerically stable, if a step size ∆x = 0.25 in space is provided?