Error analysis of a pressure correction method with explicit time stepping

TL;DR

This paper provides a theoretical error analysis of pressure correction methods with explicit time stepping for incompressible fluid simulation, highlighting stability conditions and efficiency benefits.

Contribution

It offers the first error analysis for explicit pressure correction schemes, establishing stability conditions and demonstrating their asymptotic equivalence to implicit methods.

Findings

Explicit scheme stability depends on a CFL condition.

Explicit and implicit methods show similar asymptotic behavior under CFL constraints.

Explicit methods enable efficient implementation on parallel hardware.

Abstract

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 e.g. 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 will demonstrate that the fully discrete method can be expressed 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 available literature 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 to which extend 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 demonstrate the same asymptotic behavior as the implicit variant when the CFL condition is satisfied.