ABSTRACT
The final results presented for the Crank-Nicolson method are a parametric study in which the value of T(0.4,5.0) is calculated using values of Lli = 0.1, 0.05, 0.025, and 0.0125 cm, for values of d = 0.5, 1.0, 2.0, and 5.0. The value of f"..( for each solution is determined by the specified values of Lli and d. The exact solution is 1'(0.4,5, 0) = 46.1255 C. Results are presented in Table 10.6. The truncation error of the Crank-Nicolson method is 0(f"..(2) + 0(Lli2). For a given value of d, f"..( = d fu2 /IY.. Thus, as Lli is successively halved, f"..t is quartered. Consequently, the 0(f"..(2) term should decrease by a factor of approximately 16 and the 0(Lli2) term should decrease by a factor of approximately 4 as Lli is halved for a constant value of d. The results presented in Table 10.6 show that the total error decreases by a factor of approximately 4, indicating that the 0(fu2) term is the dominant error term.
