Numerical solutions of a non‐linear density current: A benchmark solution and comparisons

Abstract

SUMMARYA comparison between solutions from simulations of a non-linear density current test problem was made in order to study the behaviour of a variety of numerical methods. The test problem was diffusion-limited so that a grid-converged reference solution could be generated using high spatial resolution. Solutions of the test problem using several different resolutions were computed by the participants of the 'Workshop on Numerical Methods for Solving Nonlinear Flow Problems', which was held on 11-13 September 1990 at the National Center for Supercomputing Applications (NCSA). In general, it was found that when the flow was adequately resolved, all of the numerical schemes produced solutions that contained the basic physics as well as most of the flow detail of the reference solution. However, when the flow was marginally resolved, there were significant differences between the solutions produced by the various models. Finally, when the flow was poorly resolved, none of the models performed very well. While higher-order and spectral-type schemes performed best for adequately and marginally resolved flow, solutions made with these schemes were virtually unusable for poorly resolved flow. In contrast, the monotonic schemes provided the most coherent and smooth solutions for poorly resolved flow, however with noticeable amplitude and phase speed errors, even at finer resolutions.

A regularization approach for a vertical‐slice model and semi‐Lagrangian Störmer–Verlet time stepping

“…'linear' regularization). The results from Figure 4 are in excellent agreement with the reference solutions provided in Straka et al (1993). Even the simulation with a low resolution of x = z = 200 m yields a reasonable approximation.…”

“…This test case has been very well documented by Straka et al (1993). We implemented the regularized equations with a constant (reference) potential temperature θ = 300 K, a constant speed of sound c s = 347 m s −1 , N = 0, and with (45) further simplified to…”

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

“…The first test case considered, a rising thermal bubble, gauges the cloud resolving capabilities of NUMA with regard to capturing fine-scale structures [14,13]. Mesoscale or limited-area modeling capabilities are tested using a density current problem first proposed by Straka et al [37]. Finally, global scale weather and climate simulation capabilities are tested with an acoustic wave propagation problem proposed in Tomita and Satoh [38].…”

“…To test the mesoscale modeling capabilities of NUMA, we use the density current problem first proposed in [37]. This test case concerns the evolution of a cold bubble in a neutrally stratified atmosphere of constant potential temperature θ 0 .…”

“…The implementations are validated with several examples that are representative of typical atmospheric dynamics. Cloud resolving capabilities are tested with a rising thermal bubble problem [14], mesoscale modeling capabilities with a density current problem [37,13] and a global scale capability is tested with an acoustic wave propagation problem run on the entire planet [38].…”