Error analysis of a pressure correction method with explicit time stepping

TL;DR

The paper analyzes error propagation for both implicit and explicit pressure-correction schemes solving the time-dependent incompressible Navier–Stokes equations via a fully discrete finite element framework. It derives stability and convergence results, establishing a CFL-type condition for the explicit scheme and showing that, under this condition, the explicit method matches the implicit method’s asymptotic behavior. A practical variant of the explicit scheme uses matrix–vector representations of the nonlinear term to avoid system reassembly, enabling efficient GPU-ready implementations. The work extends Guermond-based error analysis to the explicit setting and to the approximated nonlinear term, providing rigorous $L^2$ and $H^1$ error bounds for velocity and pressure. Numerical experiments corroborate the theoretical rates and illustrate the impact of temporal and spatial refinement as well as CFL constraints on convergence.

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.

Figures (4)

• Figure 1: Errors of the predictor velocity in the $L^2(0,T;L^2)$-norm (left) and the $L^2(0,T;H^1)$-norm (right) under time-refinement (see Tables \ref{['tab:L2L2_k']} and \ref{['tab:L2H1_k']}).
• Figure 2: Errors of the pressure in the $L^2(0,T;L^2)$-norm under time-refinement (see Table \ref{['tab:L2L2_pres_k']})
• Figure 3: Errors of the predictor velocity in the $L^2(0,T;L^2)$-norm (left) and $L^2(0,T;H^1)$-norm (right) under spatial refinement (see Tables \ref{['tab:L2L2_h']}) and \ref{['tab:L2H1_h']}.
• Figure 4: Errors of the pressure in the $L^2(0,T;L^2)$-norm under spatial refinement (see Table \ref{['tab:L2L2_pres_h']})

Theorems & Definitions (22)

• Lemma 1: Estimating the nonlinearity
• proof
• Lemma 2: Grönwall's Lemma
• Lemma 3: Projection error
• Lemma 4: Stability of the projection
• Lemma 6: Error propagation for the implicit pressure correction scheme
• Lemma 7: Approximation error estimate for the implicit pressure correction scheme
• proof
• Theorem 8: Error estimate for the implicit pressure correction scheme
• Lemma 9: Approximation error estimate for the explicit pressure correction scheme