“…Additional schemes on, and near the boundaries, at j = 0, 1, and 2, (and at j = N, N − 1, and N − 2) where the stencil protrudes the domain, are required. In this work, we use the fourth-order (second-order at j = 0 for numerical stability) optimized pentadiagonal schemes proposed by Kim and Lee [25], represented as follows: (12) (13) (14) for j = 0 (boundary node), 1, and 2, respectively. The coefficients in (12)-(14) are obtained, as described in [24], by minimizing over an optimization range of wave numbers, dispersion and, for the non-centered structure of (12)-(14) also dissipation errors.…”