ABSTRACT
The first-order time derivative It is determined directly from the partial differential equation:
- -j; = -U/t (11.33) The second-order time derivative It/ is determined by differentiating the partial differential equation with respect to time. Thus,
(11.34)
Note that this procedure does not work for a nonlinear PDE where u = u(f). Substituting Eqs. (11.33) and (11.34) into Eq. (11.32) yields
Truncating the remainder term, and approximating the two spatial derivativesfxl;' and/ul;' by second-order centered-difference approximations, Eqs. (11.20) and (10.23), respectively, gives
Ji Ji U 2 Lll: 2U Lll:2 (11.36)
OJ ~ EO ~i-1,n) (i,n) (i+1,n) Figure 11.12 The Lax-Wendroff one-step method stencil.
