Numerical Solution of the Steady-State Navier-Stokes Equations using Empirical Interpolation Methods
For computational scientists solving parameterized nonlinear PDEs, this work provides a method to drastically speed up reduced-order models for Navier-Stokes equations, though it is an incremental extension of existing empirical interpolation techniques to a specific application.
The paper demonstrates that empirical interpolation methods can significantly reduce the computational cost of solving parameterized steady-state Navier-Stokes equations, achieving online costs independent of the spatial dimension N, and shows that iterative solvers further reduce costs compared to direct methods.
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.