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c. Show that the eigenvalues λ; of A are real and that it holds A; <0.

For the integration in x-direction of (4) we take the 0-method, given by

Un+1 = Un + Ax[(1 - 0) (Aun +rn) + 0(Aun+1+Fn+1)], 0 = [0,1].

(5)

In here Az is the step size in z-direction.

d. Determine the order of the local truncation error of (5).

We choose 0 = ³.

e. Show that (5) is unconditionally stable for 0= . Is the method super-stable?

f. What is the order of the global discretization error of (5) for 0 = ?

The absorption of the gas is determined by a =y=1.

g. Give a first-order and a second-order accurate (one-sided) finite-difference formula for the computa-

tion of a. The corresponding numerical approximations of a are denoted by a₁ and 02.

h. Choose Ay = 0.05, Az = 0.02 and = 1. Plot in a single graph the numerical solution (as a

function of y) for x =nAz with n = 5, 10, 15, 20, 25. Make tables of a₁ and 0₂. Discuss the results.

i. Choose Az = 0.02, v = 1 and define L = 20Ar. We investigate the accuracy in a at x = L. Choose

Ay = 0.1, 0.05, 0.025. Make tables of a₁ and a₂ for x = L. Discuss the accuracy of a₁ and ₂2.

For this, assume that the global discretization error can be written as CAy". Estimate C and p.

Discuss the results.

j. Add the software you wrote as appendix/appendices to your report.

Fig: 1