Partially implicit finite difference scheme for calculating dynamic pressure in a terrain-following coordinate non-hydrostatic ocean model

•The PIFD scheme of non-hydrostatic calculation was proposed.•The PIFD scheme was validated by five test cases.•The PIFD scheme can reduce ∼50% computational cost relative to the conventional fully-implicit scheme.

To improve the efficiency of the terrain-following σ-coordinate non-hydrostatic ocean model, a partially implicit finite difference (PIFD) scheme is proposed. By using explicit terms instead of implicit terms to discretize the parts of the vertical dynamic pressure gradient derived from the σ-coordinate transformation, the coefficient matrix of the discrete Poisson equation that the dynamic pressure satisfies can be simplified from 15 diagonals to 7 diagonals. The PIFD scheme is shown to run stably when it is applied to simulate five benchmark cases, namely, a standing wave in a basin, a surface solitary wave, a lock-exchange problem, a periodic wave over a bar and a tidally induced internal wave. Compared with the conventional fully implicit finite difference (FIFD) scheme, the PIFD scheme produces simulation results of equivalent accuracy at only 40–60% of the computational cost. The PIFD scheme demonstrates strong applicability and can be easily implemented in σ-coordinate ocean models.