Propagation of internal gravity waves: Optimal vertical discretisation and grid design based on numerical error analysis

Abstract Focusing on the dynamics of internal gravity waves, we introduce a general framework for the vertical discretisation of the stratified linearised primitive equations. We derive a discrete formulation that preserves energy conservation and leads to a consistent discrete Sturm–Liouville problem. A ‐indexed family of discretisations is introduced and a perturbation analysis provides an explicit expression for the leading‐order eigenvalue errors. We show that an appropriate choice of parameter reduces the leading‐order eigenvalue error significantly and, for uniform grids, makes it vanish, yielding fourth‐order instead of second‐order convergence. An explicit optimal vertical grid is also derived for a single eigenmode, and the extension to grids optimised for multiple modes is discussed.