Truncation Error of Explicit Scheme

With the Taylor expansions given by Eqs. (4.41) and (4.42) and similar expansions in time, we employ the forward difference in time and the central difference in space to get 

Equation (4.70) indicates that the governing partial differential equation (PDE) equals the finite-difference approximation (FDA) to within a truncation error (TE) which is of 0[Af, (Ax)2]. Clearly, the first-order forward difference has only first-order accuracy [recall Eq. (4.43)]. However, differentiating BT/Bt = ud2T/dx2 twice with respect to x and once with respect to f, respectively, shows that the first two terms of TE can be combined into one, 

for the particular choice of F — 1/6 the explidt scheme has TE of 0[(Ar)2, (Ax)4], which assures ahigher accuracy. However, the choice At = (Ax)2/(6a) implies time steps too small to be of practical interest in most cases. Irrespective of the size of the time step relative to that of the space step, TE 0, hence FDA — PDE, as Ai - 0 and Ax — 0, which shows the discrete formulation to be consistent 

A flat plate with thickness l = 10 cm and diffusivity or = 1 x 10 4 m2/s is initially at temperature of 0 °C. One surface of the plate is kept at a temperature of 0 °C while the other is suddenly raised to 100 °C. We wish to determine the transient temperature distribution within the plate by using an explicit numerical scheme. 

we let At — 1 s, i.e., F = 0.25, to ensure the stability. The FORTRAN program EX4-7.F solves this problem. Table 4.9 gives the iteration results.  

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