On the performance of high resolution non-oscillating advection schemes in the context of the flow generated by a mixed region in a stratified fluid

The two-dimensional flow generated by a local density perturbation (fully mixed region) in stratified fluid is considered. In order to describe accurately the sharp discontinuity in density at the edge of the mixed region, monotone schemes of high order of approximation are required. Although a great variety of methods have been developed during the last decades, there remains the question of which method is the best. This present paper deals with the numerical treatment of the advective terms in the Navier-Stokes equations in the Oberbeck-Boussinesq approximation. Comparisons are made between the upwind scheme, flux-limiter schemes, namely Minmod, Superbee, van Leer and monotonized centred (MC), monotone adaptive stencil schemes, namely ENO3 and SMIF, and the weighted stencil scheme WENO5. We used the laboratory experimental data of Wu as a benchmark test to compare the performance of the various numerical approaches. We found that the flux limiter schemes have the smallest numerical diffusion. On the other hand, the WENO5 scheme describes the variation of the width of the collapsing region over time most accurately. All considered schemes give realistic patterns of internal gravity waves generated by the collapsing region.