Evaluation of Selected Finite-Difference and Finite-Volume Approaches to Rotational Shallow-Water Flow

Abstract

The shallow-water equations in a rotating frame of reference are important for capturing geophysical flows in the ocean. In this paper, we examine and compare two traditional finite-difference schemes and two modern finite-volume schemes for simulating these equations. We evaluate how well they capture the relevant physics for problems such as storm surge and drift trajectory modelling, and the schemes are put through a set of six test cases. The results are presented in a systematic manner through several tables, and we compare the qualitative and quantitative performance from a cost-benefit perspective. Of the four schemes, one of the traditional finitedifference schemes performs best in cases dominated by geostrophic balance, and one of the modern finite-volume schemes is superior for capturing gravity-driven motion. The traditional finite-difference schemes are significantly faster computationally than the modern finite-volume schemes.

Real-World Oceanographic Simulations on the GPU using a Two-Dimensional Finite-Volume Scheme

“…Figure 3 demonstrates how the angle to north plays an important role in the generation of planetary Rossby waves, caused by variations in the Coriolis force using (39). The figure shows the resulting η after long simulations initialized with a rotating Gaussian bump in geostrophic balance in the middle of the domain, similar to the case presented in Holm et al [27].…”

“…using (21). Here, we see that H cancels immediately, whereas η cancels by the use of (27). To make the last terms cancel, we need to set f −1, j v −1, j = − f 0, j v 0, j , which means that in addition to (27), we need to enforce…”

“…We have previously evaluated the standard CDKLM scheme on several test cases which target oceanographic dynamics relevant for the shallow-water equations when implemented on a Cartesian grid [27]. In this work, however, we have reformulated the scheme in terms of η instead of h, added several new source terms, and enabled varying latitude and north vector.…”

GPU Computing with Python: Performance, Energy Efficiency and Usability

“…Class The schemes are used for simulating real-world ocean currents, and two of them have been used operationally in the early days of computational oceanography. All three schemes are essentially stencil operations with an increasing level of complexity, and their details are summarized in Holm et al [8].…”

“…Source terms represent varying bathymetry and the Coriolis forces, which takes into account that we solve the equations on a rotating sphere. The equations can be used to model gravitational waves, in which the governing motion is horizontal such as e.g., the ocean [7,8]. An oceanographic simulation scenario using our simulator is shown in Figure 2.…”