An Unsuspected Boundary-Induced Temporal Computational Mode in a Two-Time-Level Discretization

Abstract

Normal-mode analyses are applied to various discrete forms of the one-dimensional, linearized, vertical acoustic equations in a height-based coordinate. First, the temporally discrete, spatially continuous equations are considered and the normal modes for a bounded system are compared to those of an unbounded system. Despite the use of a two-time-level discretization, a computational mode is found in the bounded case that is absent in the unbounded case. Second, the complete temporally and spatially discrete bounded system is considered and the normal modes and associated dispersion relation are derived. No computational modes are found. However, under certain limiting conditions, the temporal discretization leads to a distortion of the physical modes so that they resemble the computational mode of the spatially continuous bounded system. Implications for analyses based on spatially continuous equation sets are discussed.