[Paper Review] An Artificial Compressibility Ensemble Timestepping Algorithm for Flow Problems

Joseph A. Fiordilino, Michael McLaughlin|arXiv (Cornell University)|Dec 18, 2017

Fluid Dynamics and Turbulent Flows5 references3 citations

TL;DR

This paper introduces an artificial compressibility ensemble (ACE) timestepping algorithm that decouples velocity, pressure, and temperature solves via artificial compressibility and IMEX splitting, enabling first-order convergence and nonlinear energy stability under a CFL-type condition. The method reduces computational cost and storage by reusing a constant coefficient matrix across ensemble members, significantly improving efficiency for ensemble flow simulations in laminar regimes.

ABSTRACT

Ensemble calculations are essential for systems with uncertain data but require substantial increase in computational resources. This increase severely limits ensemble size. To reach beyond current limits, we present a first-order artificial compressibility ensemble algorithm. This algorithm effectively decouples the velocity and pressure solve via artificial compression, thereby reducing computational complexity and execution time. Further reductions in storage and computation time are achieved via a splitting of the convective term. Nonlinear energy stability and first-order convergence of the method are proven under a CFL-type condition involving fluctuations of the velocity. Numerical tests are provided which confirm the theoretical analyses and illustrate the value of ensemble calculations.

Motivation & Objective

• To address the high computational cost and memory demands of large-scale ensemble simulations in fluid dynamics with uncertain parameters.
• To develop a timestepping algorithm that reduces complexity and execution time while maintaining accuracy for ensemble flow problems.
• To enable larger ensemble sizes without sacrificing mesh resolution by decoupling velocity, pressure, and temperature solves through artificial compressibility.
• To prove nonlinear energy stability and first-order convergence under a CFL-type condition involving velocity fluctuations.
• To demonstrate the method's effectiveness through numerical experiments validating theoretical analysis.

Proposed method

• The method employs an IMEX (implicit-explicit) time discretization to decouple nonlinear convective terms, enabling a shared coefficient matrix across ensemble members.
• Artificial compressibility is introduced by adding a penalty term $ \epsilon p_t $ to the continuity equation, transforming the saddle-point problem into a convection-diffusion problem with grad-div stabilization.
• The velocity, pressure, and temperature equations are fully decoupled: velocity and temperature solve convection-diffusion systems, while pressure is updated algebraically from divergence of velocity.
• The convective terms are split into ensemble mean and fluctuating components: $ u \cdot \nabla u = \langle u \rangle \cdot \nabla u + u' \cdot \nabla u $, ensuring matrix invariance across ensemble members.
• A CFL-type stability condition is derived, depending on fluctuations in the velocity field, which limits applicability to laminar flows.
• The algorithm is implemented in a semi-implicit, first-order time-stepping framework with spatial discretization applied to the resulting linear systems.

Experimental results

Research questions

• RQ1 Can artificial compressibility be effectively combined with ensemble timestepping to reduce computational complexity and storage in flow simulations?
• RQ2 Does the proposed ACE algorithm achieve first-order convergence and nonlinear energy stability under a velocity-fluctuation-dependent CFL condition?
• RQ3 To what extent does the decoupling of velocity, pressure, and temperature solves reduce computational cost and enable larger ensemble sizes?
• RQ4 How does the method perform in terms of accuracy and predictability for varying Rayleigh numbers and initial condition perturbations?
• RQ5 Why does the average effective Lyapunov exponent for pressure become negative while variance increases, and how does this affect predictability?

Key findings

• First-order convergence is numerically confirmed for velocity, temperature, and pressure, with observed rates matching theoretical predictions.
• Pressure error converges at a half-order higher rate than predicted, indicating improved accuracy in pressure approximation.
• The ensemble average remains close to the unperturbed solution, while perturbed solutions diverge significantly with increasing Rayleigh number, especially in velocity and pressure.
• Variance increases with Rayleigh number, indicating growing uncertainty in velocity and pressure predictions, while temperature remains relatively reliable.
• The average effective Lyapunov exponent increases with time and Rayleigh number, indicating decreasing predictability for velocity and temperature, but becomes negative for pressure, suggesting increasing predictability over time.
• The $ \delta $-predictability horizon decreases with increasing Rayleigh number, confirming reduced predictability in turbulent regimes, with the smallest horizon observed at $ Ra = 10^4 $.