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This is an unstable mode if µ(1 − 2θ) > 12 . 78) This includes the earlier explicit case, θ = 0: and it also shows that the fully implicit scheme with θ = 1 is not unstable for any value of µ. Indeed no scheme with θ ≥ 12 is unstable for any µ. 78) is not satisfied, we have |λ(k)| ≤ 1 for every mode k, so that no mode grows at all and the scheme 28 Parabolic equations in one space variable is stable. 79)   1 when 2 ≤ θ ≤ 1, stable for all µ. The two cases are often referred to as conditional and unconditional stability respectively.

19). Thus it takes about twice as long for each time step. The importance of the implicit method is, of course, that the time steps can be much larger, for, as we shall see, there is no longer any stability restriction on ∆t. We shall give a proof of the convergence of this implicit scheme in the next section, as a particular case of a more general method. 7. We construct a solution of the difference equations for Fourier modes of the same form as before, Ujn = (λ)n eik(j∆x) . 73) 26 Parabolic equations in one space variable which shows that λ= 1 .

6: and any engineering client for our computed results would be rather dismayed if they did not possess this property. We generalise that result by the following theorem. 90) and Umax := max U0m , 0 ≤ m ≤ n; Uj0 , 0 ≤ j ≤ J; UJm , 0 ≤ m ≤ n . 75) with consistent initial and Dirichlet boundary data converge uniformly on [0, 1] × [0, tF ] if the inin+1/2 tial data are smooth enough for the truncation error Tj to tend to zero along the refinement path uniformly in this domain. 75) in the form n+1 n+1 n n (1 + 2θµ)Ujn+1 = θµ Uj−1 + (1 − θ)µ Uj−1 + Uj+1 + Uj+1 + [1 − 2(1 − θ)µ] Ujn .