/'+

The modified differential equation (MDE) corresponding to Eq. (10.67) is it = alxx + ~ itt dt - V;1I dt2 + ... + -&. alxxxx M + 3~O alxxxxxx Ax4 + ... (10.68)

As tJ.t ~ 0 and Ax ~ 0, all of the truncation error terms go to zero, and Eq. (10.68) approaches it = alxx ' Consequently, Eq. (10.67) is consistent with the diffusion equation. The truncation error is O(dt) + 0(Ax2 ). From a von Neumann stability analysis, the amplification factor G is

IG=------ 1 + 2d(1 - cos ()

The term (1 - cos () is greater than or equal to zero for all values of () = (km Ax). Consequently, the denominator ofEq. (10.69) is always::: 1. Thus, IGI S 1 for all positive values of d, and Eq. (10.67) is unconditionally stable. The BTCS approximation of the diffusion equation is consistent and unconditionally stable. Consequently, by the Lax Equivalence Theorem, the BTCS method is convergent.