Stability of Two Conservative, High-Order Fluid-Fluid Coupling Methods (Advances in Applied Mathematics and Mechanics, 2021)
This paper investigates two methods of coupling fluids across an interface, motivated by air-sea interaction in application codes. One method is for sequential configurations, where the air code module in invoked over some time interval prior to the sea module. The other method is for concurrent setups, in which the air and sea modules run in parallel. The focus is the temporal representation of air-sea fluxes. The methods we study conserve moments of the fluxes, with an arbitrary order of accuracy possible in time. Different step sizes are allowed for the two fluid codes. An $a$ $posteriori$ stability indicator is defined, which can be computed efficiently on-the-fly over each coupling interval. For a model of two coupled fluids with natural heat convection, using finite elements in space, we prove the sufficiency of our stability indicator. Under certain conditions, we also prove that stability can be enforced by iteration when the coupling interval is small enough. In particular, for solutions in a certain class, we show that the step size scaling is no worse than $\mathcal{O}(h)$ in three dimensions of space, where $h$ is a mesh parameter. This is a sharper result than what has been shown previously for related algorithms with finite element methods. Computational examples illustrate the behavior of the algorithms under a wide variety of configurations.