[Paper Review] An efficient algorithm for simulating ensembles of parameterized flow problems

Max Gunzburger, Nan Jiang+1 more|arXiv (Cornell University)|May 25, 2017

Fluid Dynamics and Turbulent Flows16 references3 citations

TL;DR

This paper proposes a novel semi-implicit ensemble algorithm for simulating multiple parameterized incompressible Navier-Stokes flows with varying viscosity coefficients, initial conditions, and body forces. By constructing a single linear system with multiple right-hand sides using a common coefficient matrix, the method achieves significant computational savings while maintaining first-order temporal accuracy and optimal spatial convergence, with rigorous stability and error estimates under time-step and parameter-deviation constraints.

ABSTRACT

Many applications of computational fluid dynamics require multiple simulations of a flow under different input conditions. In this paper, a numerical algorithm is developed to efficiently determine a set of such simulations in which the individually independent members of the set are subject to different viscosity coefficients, initial conditions, and/or body forces. The proposed scheme applied to the flow ensemble leads to need to solve a single linear system with multiple right-hand sides, and thus is computationally more efficient than solving for all the simulations separately. We show that the scheme is nonlinearly and long-term stable under certain conditions on the time-step size and a parameter deviation ratio. Rigorous numerical error estimate shows the scheme is of first-order accuracy in time and optimally accurate in space. Several numerical experiments are presented to illustrate the theoretical results.

Motivation & Objective

• To develop an efficient numerical algorithm for simulating ensembles of parameterized incompressible flow problems with distinct viscosity coefficients, initial conditions, and body forces.
• To reduce computational cost by solving a single linear system with multiple right-hand sides instead of solving separate systems for each ensemble member.
• To establish rigorous nonlinear and long-term stability under time-step and parameter deviation ratio constraints.
• To prove first-order temporal accuracy and optimal spatial accuracy via a comprehensive error estimate.
• To validate the theoretical findings through numerical experiments on benchmark flow problems.

Proposed method

• A semi-implicit temporal discretization is applied to the Navier-Stokes equations, where the nonlinear convection term is treated explicitly and the viscous and pressure terms implicitly.
• A novel spatial discretization is introduced that ensures the same coefficient matrix is used across all ensemble members, regardless of differing viscosities, initial conditions, or forcing functions.
• The resulting linear system at each time step has a single coefficient matrix and J right-hand sides, one per ensemble member, enabling efficient solution via block solvers.
• Stability analysis is conducted using a discrete energy method, incorporating a modified discrete Gronwall inequality to control error propagation.
• Error estimates are derived in multiple norms, including $L^2$, $H^1$, and $L^ ho$-type norms, with bounds depending on viscosity deviation, time-step size, and mesh size.
• The method is extended to handle variable viscosity by introducing a reference viscosity $\overline{\nu}$ and analyzing the deviation $|\nu_j - \overline{\nu}|$ in the stability and error bounds.

Experimental results

Research questions

• RQ1 Can a single linear system with multiple right-hand sides be constructed for ensemble simulations of Navier-Stokes flows with varying viscosities, initial conditions, and body forces?
• RQ2 What conditions on the time-step size and viscosity deviation ratio ensure nonlinear and long-term stability of the ensemble scheme?
• RQ3 What is the temporal and spatial convergence rate of the proposed ensemble algorithm, and is it optimal?
• RQ4 How does the error bound scale with respect to viscosity deviation, time-step, and mesh size?
• RQ5 Can the proposed method significantly reduce computational cost compared to solving each simulation independently?

Key findings

• The proposed algorithm achieves first-order accuracy in time and optimal $O(h^k)$ convergence in space for velocity and pressure approximations, where $k$ is the polynomial degree of the finite element space.
• The method is unconditionally nonlinearly stable and long-term stable under the condition that the time-step size and viscosity deviation ratio satisfy a specific bound involving $\overline{\nu}$ and $|\nu_j - \overline{\nu}|$.
• The error bound scales as $O(\Delta t^2 + h^{2k})$, with constants depending on $\overline{\nu}^{-1}$, $\|\nabla u_{j,t}\|_{L^2}$, and higher-order derivatives of the solution.
• The computational cost is reduced from solving $J$ separate linear systems per time step to solving only one system with $J$ right-hand sides, significantly improving efficiency.
• Numerical experiments confirm the theoretical error estimates and demonstrate the method's robustness and efficiency across various parameter variations.
• The method maintains stability and accuracy even when viscosity coefficients vary widely across ensemble members, provided the deviation ratio condition is satisfied.