Error Analysis of a Pressure-Correction Method With Explicit Time-Stepping

The pressure-correction method is a well-established approach for simulating unsteady, incompressible fluids. It is well-known that implicit discretization of the time derivative in the momentum equation, for example, using a backward differentiation formula with explicit handling of the nonlinear term, results in a conditionally stable method. In certain scenarios, employing explicit time integration in the momentum equation can be advantageous, as it avoids the need to solve for a system matrix involving each differential operator. Additionally, we demonstrate that the fully discrete method can be written in the form of simple matrix-vector multiplications, allowing for efficient implementation on modern and highly parallel acceleration hardware. Despite being a common practice in various commercial codes, there is currently no literature available on error analysis for this scenario. In this work, we conduct a theoretical analysis of both implicit and two explicit variants of the pressure-correction method in a fully discrete setting. We demonstrate the extent to which the presented implicit and explicit methods exhibit conditional stability. Furthermore, we establish a Courant–Friedrichs–Lewy (CFL) type condition for the explicit scheme and show that the explicit variant demonstrates the same asymptotic behavior as the implicit variant when the CFL condition is satisfied.