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where the fourth-order dissipation term is added to prevent instability and it is recalculated in every stage while marching. This model equation is chosen to fit with the use of the effective third-order dissipative term that is implemented throughout smooth region. The multistage schemes designed are considered the artificial viscosity coefficient IJL of ^, -^, and ^. By using the van Leer optimization, the parameters of the optimal schemes for given central differencing, without use of residual smoothing, are shown in Table 1. In the table, v denotes the CourantFriedrichs-Lewy (CFL) number, otk is the Ath multistage coefficient, and |P|max is the equalized minimal maximum of the amplification factor of ^-stage scheme in the range [n/2, n]. The contours of the amplification factor of the optimal four-stage scheme for the central spatial differencing with IJL = -^, together with the locus of the Fourier transform z(p) of the spatial operator (dashed line) are shown in Fig. la. The contour levels for the magnitude of the amplification factor are |P| = 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.05, and 0.01. Note that the CFL number achieved in an £-stage scheme is considerably lower than the maximum stable CFL number for that £-stage scheme, which makes the optimal multistage