# A Semi-Explicit Multi-Step Method for Solving Incompressible Navier-Stokes Equations
## 1 Introduction  
The solution of incompressible flow problems arising in real-life engineering applications calls for developing accurate, yet efficient schemes, as the associated computational time becomes a decisive factor for their use. Among the many methods available, one class of techniques commonly implemented is the well-known fractional step method. The method was originally introduced in the works of [PERSON] [1] and [PERSON] [2] for incompressible Navier-Stokes equations (independently, a similar methodology was developed and thoroughly studied by [PERSON] [3, 4]). Basically, the method allows one to decouple the primary variables of the governing equations (the velocity and the pressure) and replaces the \"monolithic\" solution of the original coupled system by a series of solution steps, ensuring that only one variable (either the velocity or the pressure) is unknown at each step. First, the momentum equation is solved for an intermediate, non-solenoidal, velocity removing the dependence on the unknown pressure. Afterwards, the pressure is found, and the velocity is corrected fulfilling the incompressibility constraint. It has been shown that the fractional step method can be viewed as a block LU decomposition of the original monolithic equations system [5]. The method is very popular due to the resulting computational efficiency (see, e.g., [6, 7, 8]). The accuracy of the method was addressed in [9]. Error estimates and pressure stability issues of the fractional step method have been analyzed in [10]. An exhaustive review of different implementation versions of the method can be found in [11]. A method that unifies the artificial compressibility approach and a fractional step method can be consulted in [12]. The fractional step methodology has been also applied to compressible flows modeling (see, e.g., [13; 14]). The peculiarities of applying the fractional step method to multiphase flows with surface tension effects have been highlighted in [15]. While the majority of works on fractional step methods had been carried out considering fixed-grid (Eulerian) approaches, some recent works have analyzed the implications of applying the fractional step method to moving grid (Lagrangian) methods [16; 17]. The fractional step method has been also applied for modeling incompressible solids [18]. In the context of incompressible flow modeling, the fractional step method is often applied in conjunction with implicit time integration schemes. This is the case for all the above-mentioned works. Purely explicit schemes cannot be applied to incompressible Navier-Stokes equations due to the implicit nature of the pressure. On the other hand, an appealing possibility consists of developing a semi-explicit scheme, where velocity is integrated explicitly, while pressure is treated implicitly. A particularly appealing option relies on adopting an accurate high order methodology such as the Runge-Kutta time integration family. This has been followed in several works, e.g., [19; 20; 21; 22; 23; 24; 25; 26]. In the context of finite volume methods, this was followed, e.g., in [21; 22]. In [21], the accuracy of different versions of Runge-Kutta schemes is analyzed. The authors derive a predictor-corrector scheme, which is very similar to a fractional step one; however, no spitting error is committed as pressure is computed at every sub-step of the Runge-Kutta scheme. In [22], stability bounds of different fractional step-based schemes (including the ones using Runge-Kutta integration) are analyzed for high Reynolds number flows. Several authors developed semi-explicit Runge-Kutta-based methods in the context of the finite difference framework. In [22], a third-order Runge-Kutta scheme for the convective term combined with a Crank-Nicholson integration for the viscous term was proposed. In [23], a fourth-order Runge-Kutta scheme equipped with the solution of the pressure Poisson equation at each sub-step is proposed (similar to [21]).