special paper. Numerical Method Conventional approaches (e.g., 1988; Singh et al. 1988; Chang and Zhang 2010) start the calculation from an initial time with a given step for which the increments and are t0 t, ΔH, Δz, ΔV obtained iteratively. By examining these equations, it can be found that once V is given, the solutions to and can be obtained by straightforward calculations without need of iteration. ΔH, Δz, Δt Therefore, a new approach is herein proposed that starts from an initial velocity V0 with an interval of The formulations and procedures of this new approach are described in the following subsections. ΔV. Formulations At a velocity step from to the average velocityis V0 V0+V, (27) The average values of and are H z (28) (29) From Eq. (5), the average velocity can be found by (30) (31) Once is given, is obtained with Eq. (27) and then is obtained by reformulating Eq. (31) as V s (32) Eqs. (10) and (14) or (16) can be expressed in finite difference forms, respectively, as (33) (34) where is determined by Eq. (13), in which and are replaced by their average values and V h determined by Eqs. (27) and (2), respectively. Eliminating and in Eqs. (33) and (34), one obtains t H (35) where (36) (37) (38) (39)
special paper. Numerical Method Conventional approaches (e.g., 1988; Singh et al. 1988; Chang and Zhang 2010) start the calculation from an initial time t0 with a given step t, for which the increments ΔH, Δz, and ΔV are obtained iteratively. By examining these equations, it can be found that once V is given, the solutions to ΔH, Δz, and Δt can be obtained by straightforward calculations without need of iteration. Therefore, a new approach is herein proposed that starts from an initial velocity V0 with an interval of ΔV. The formulations and procedures of this new approach are described in the following subsections. Formulations At a velocity step from V0 to V0+V, the average velocityis (27) The average values of H and z are (28) (29) From Eq. (5), the average velocity can be found by (30) (31) Once V is given, is obtained with Eq. (27) and then s is obtained by reformulating Eq. (31) as (32) Eqs. (10) and (14) or (16) can be expressed in finite difference forms, respectively, as (33) (34) where is determined by Eq. (13), in which V and h are replaced by their average values and determined by Eqs. (27) and (2), respectively. Eliminating t and H in Eqs. (33) and (34), one obtains (35) where (36) (37) (38) (39)
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