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This study proposes an algorithm for modeling compressible flows in spherical shells in nearly incompressible and weakly compressible regimes based on an implicit direction splitting approach. The method retains theoretically expected convergence rates and remains stable for extremely small values of the characteristic Mach number. The staggered spatial discretization on the MAC stencil, commonly used in numerical methods for incompressible Navier-Stokes equations, was found to be convenient for the discretization of the compressible Navier-Stokes equations written in the non-conservative form in terms of the primitive variables. This approach helped to avoid the high-frequency oscillations without any artificial stabilization terms. Nonlinear Picard iterations with the splitting error reduction were also implemented to allow one to obtain a solution of the fully nonlinear system of equations. These results, alongside excellent parallel performance, prove the viability of the direction splitting approach in large-scale high-resolution high-performance simulations of atmospheric and oceanic flows.

This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical coordinate transforms. This approach allows to avoid the singularities at the poles and keeps the grid size relatively uniform. The downside is that the spherical shell is subdivided into two equally sized, overlapping subdomains that requires the use of Schwarz-type iterations. The temporal second order accuracy is achieved via an Artificial Compressibility (AC) scheme with bootstrapping. The spatial discretization is based on second order finite differences on the Marker-And-Cell (MAC) stencil. The entire scheme is implemented in parallel using a domain decomposition iteration, and a direction splitting approach for the local solves. The stability, accuracy and weak scalability of the method is verified on a manufactured solution of the Navier-Stokes-Boussinesq system and on the Landau solution of the Navier-Stokes equations on the sphere.

The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows. Although many highly efficient algorithms have been proposed for simulating each type of the flows, the construction of a universal solver is known to be challenging. Extreme cases, such as incompressible and highly compressible flows, or inviscid and highly viscous flows, require different numerical treatments in order to maintain the efficiency, stability, and accuracy of the method. Linearized block implicit (LBI) factored schemes are known to provide an efficient way of solving the compressible Navier-Stokes equations implicitly, allowing us to avoid stability restrictions at low Mach number and high viscosity. However, the methods' splitting error has been shown to grow and dominate physical fluxes as the Mach number goes to zero. In this paper, a splitting error reduction technique is proposed to solve the issue. A novel finite element shock-capturing algorithm, proposed by Guermond and Popov, is reformulated in terms of finite differences, extended to the stiffened gas equation of state (SG EOS) and combined with the LBI factored scheme to stabilize the method around flow discontinuities at high Mach numbers. A novel stabilization term is proposed for low Mach number applications. The resulting algorithm is shown to be efficient in both low and high Mach number regimes. The algorithm is extended to the multicomponent case using an interface capturing strategy with surface tension as a continuous surface force. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.