Numerical Solution of the Steady-State Navier-Stokes Equations using Empirical Interpolation Methods

TL;DR

This paper demonstrates how empirical interpolation methods can significantly reduce the computational costs of solving steady-state Navier-Stokes equations in reduced-order models, especially for nonlinear problems.

Contribution

It introduces the application of empirical interpolation methods to nonlinear Navier-Stokes equations, enabling efficient reduced-order modeling with lower online computational costs.

Findings

Empirical interpolation reduces online costs for nonlinear PDEs.

Iterative solvers further decrease algebraic system solution costs.

Reduced-order models maintain accuracy with significantly less computation.

Abstract

Reduced-order modeling is an efficient approach for solving parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem at significantly lower cost, producing an accurate estimate of the solution. For nonlinear problems, however, standard methods do not achieve the desired cost savings. Empirical interpolation methods represent a modification of this methodology used for cases of nonlinear operators or nonaffine parameter dependence. These methods identify points in the discretization necessary for representing the nonlinear component of the reduced model accurately, and they incur online computational costs that are independent of the spatial dimension $N$. We will show that empirical interpolation methods can be used to significantly reduce the costs of solving parameterized versions of the Navier-Stokes equations, and that iterative solution methods can be used in place of direct methods to further reduce the costs of solving the algebraic systems arising from reduced-order models.