On well-posed energy/entropy stable boundary conditions for the rotating shallow water equations

Abstract: We derive and analyze well-posed, energy- and entropy-stable boundary conditions (BCs) for the two-dimensional linear and nonlinear rotating shallow water equations (RSWE) in vector invariant form. The focus of the study is on subcritical flows, which are commonly observed in atmospheric, oceanic, and geostrophic flow applications. We consider spatial domains with smooth boundaries and formulate both linear and nonlinear BCs using mass flux, Riemann's invariants, and Bernoulli's potential, ensuring that the resulting initial boundary value problem (IBVP) is provably entropy- and energy-stable. The linear analysis is comprehensive, providing sufficient conditions to establish the existence, uniqueness, and energy stability of solutions to the linear IBVP. For the nonlinear IBVP, which admits more general solutions, our goal is to develop nonlinear BCs that guarantee entropy stability. We introduce the concepts of linear consistency and linear stability for nonlinear IBVPs, demonstrating that if a nonlinear IBVP is both linearly consistent and linearly stable, then, for sufficiently regular initial and boundary data over a finite time interval, a unique smooth solution exists. Both the linear and nonlinear IBVPs can be efficiently solved using high-order accurate numerical methods. By employing high-order summation-by-parts operators to discretize spatial derivatives and implementing weak enforcement of BCs via penalty techniques, we develop provably energy- and entropy-stable numerical schemes on curvilinear meshes. Extensive numerical experiments are presented to verify the accuracy of the methods and to demonstrate the robustness of the proposed BCs and numerical schemes.