The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows, including nearly incompressible low-Mach number flows, flows with strong shocks, multicomponent flows with high density ratios and interfacial physics, inviscid and viscous flows, as well as flows featuring combinations of these phenomena and various interactions between them. Although many highly efficient algorithms have been proposed for simulating each type of the flows mentioned above, 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 (see e.g.[1], [2]) are known to provide an efficient way of solving the compressible Navier-Stokes equations (compressible NSEs) 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 approaches zero (see [3]). In this paper, a splitting error reduction technique is proposed to solve the issue. A shock-capturing algorithm from [4] 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. Next, the algorithm is extended to a multicomponent case using an interface capturing strategy (see e.g. [5]) with surface tension as a continuous surface force (see [6]). Special care is taken to avoid spurious oscillations of pressure and generation of artificial acoustic waves in the numerical mixture layer. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.

In this article, we present a direction splitting method, combined with a nonlinear iteration, for the compressible Navier‐Stokes equations in spherical coordinates. The method is aimed at solving the equations on the sphere, and can be used for a regional geophysical simulations as well as simulations on the entire sphere. The aim of this work was to develop a method that would work efficiently in the limit of very small to vanishing Mach numbers, and we demonstrate here, using a numerical example, that the method shows good convergence and stability at Mach numbers in the range [10−2, 10−6]. We also demonstrate the effect of some of the parameters of the model on the solution, on a common geophysical test case of a rising thermal bubble. The algorithm is particularly suitable for a massive parallel implementation, and we show below some results demonstrating its excellent weak scalability.

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