[Paper Review] A second-order time-stepping scheme for simulating ensembles of parameterized flow problems

Max Gunzburger, Nan Jiang+1 more|arXiv (Cornell University)|Jun 9, 2017

Numerical methods for differential equations7 references3 citations

TL;DR

This paper proposes a second-order accurate, ensemble-based time-stepping scheme for efficiently simulating multiple incompressible Navier-Stokes flow problems with varying parameters, initial conditions, boundary conditions, and forcing functions. By using a semi-implicit, ensemble-averaged formulation, the method solves a single linear system with multiple right-hand sides per time step, significantly reducing computational cost while maintaining second-order accuracy and proving stability and error estimates.

ABSTRACT

We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Motivation & Objective

• To address the high computational cost of simulating multiple flow simulations with varying physical parameters, initial conditions, boundary conditions, and forcing functions.
• To develop a time-accurate, second-order numerical method that maintains efficiency across diverse flow ensembles.
• To ensure stability and derive rigorous error estimates for the proposed ensemble scheme.
• To enable efficient large-scale simulations for applications such as data assimilation, reduced-order modeling, and sensitivity analysis.

Proposed method

• The method employs a second-order semi-implicit time discretization using a three-step backward difference formula for velocity and pressure updates.
• It introduces an ensemble mean velocity $ \overline{u}^n $ and a common viscosity $ \overline{\nu} $, allowing a single coefficient matrix to be used across all ensemble members.
• The formulation decouples the nonlinear convection term using a combination of ensemble-averaged and fluctuation terms, ensuring consistency and stability.
• The discrete system is solved as one linear system with $ J $ right-hand sides, where $ J $ is the number of ensemble members, enabling efficient LU or block iterative solvers.
• The method preserves the incompressibility constraint $ \nabla \cdot u_j^{n+1} = 0 $ at each time step.
• Stability and error analysis are conducted using energy estimates and the discrete Gronwall inequality, leading to optimal convergence rates in $ L^2 $ and $ H^1 $ norms.

Experimental results

Research questions

• RQ1 Can a second-order time-accurate ensemble scheme be constructed that maintains high efficiency across multiple Navier-Stokes simulations with varying parameters and inputs?
• RQ2 How can a single linear system with multiple right-hand sides be used to simulate multiple flow ensembles without sacrificing accuracy or stability?
• RQ3 What is the optimal balance between ensemble averaging and individual member correction to ensure second-order convergence?
• RQ4 What are the theoretical error bounds and stability conditions for the proposed scheme under varying viscosity and forcing functions?
• RQ5 How does the method scale in terms of computational cost compared to solving individual systems for each ensemble member?

Key findings

• The proposed scheme achieves second-order accuracy in time for both velocity and pressure approximations under appropriate regularity assumptions.
• The method ensures conditional stability under a time-step restriction dependent on the ensemble viscosity and solution regularity.
• Error estimates show optimal convergence rates: $ \mathcal{O}(h^{k+1}) $ in $ H^1 $-norm for velocity and $ \mathcal{O}(h^{s+1}) $ in $ L^2 $-norm for pressure, with $ k $ and $ s $ related to finite element polynomial degrees.
• The computational cost is reduced by solving only one linear system per time step with $ J $ right-hand sides, instead of $ J $ separate systems.
• The error bound includes terms dependent on $ |\nu_j - \overline{\nu}| $, indicating sensitivity to viscosity variation, but this is bounded by $ \overline{\nu}^{-1} $, ensuring stability for small relative differences.
• Numerical experiments confirm the theoretical error estimates and demonstrate the efficiency gains in practice, especially for large $ J $.