Figure 1 The x-t solution domain for the weighted four-point implicit scheme. See ftp site for color image.

parallel to the t axis represent locations of cross sections; they have a spacing of Ax, which need not be the same between each pair of cross sections. Those parallel to the x axis represent time lines; they have a spacing of At, which also need not be the same between successive time lines. Each point in the rectangular network can be identified by a subscript (i), which designates the x position or cross section, and a superscript (j), which designates the particular time line.

Numerical Approximations. The time derivatives are approximated by a forward-difference quotient at point (x', t') centered between the i and i + 1 lines along the x axis as shown in Figure 1, i.e.,

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