An Ensemble-Based Projection Method and Its Numerical Investigation

In many cases, partial differential equation (PDE) models involve a set of parameters whose values may vary over a wide range in application problems, such as optimization, control and uncertainty quantification. Performing multiple numerical simulations in large-scale settings often leads to tremendous demands on computational resources. Thus, the ensemble method has been developed for accelerating a sequence of numerical simulations. In this work we first consider numerical solutions of Navier-Stokes equations under different conditions and introduce the ensemblebased projection method to reduce the computational cost. In particular, we incorporate a sparse grad-div stabilization into the method as a nonzero penalty term in discretization that does not strongly enforce mass conservation, and derive the long time stability and the error estimate. Numerical tests are presented to illustrate the theoretical results. A simple way to solve the linear system generated in the ensemble method is to use a direct solver. Compared with individual simulations of the same problems, the ensemble method is more efficient because there is only one linear system needs to solve for the ensemble. However, for large-scale problems, iterative linear solvers have to be used. Therefore, in the second part of this work we investigate numerical performance of the ensemble method with block iterative solvers for two typical evolution problems: the heat equation and the Navier-Stokes equations. Numerical results are provided to demonstrate the effectiveness and efficiency of the ensemble method when working together with the block iterative solvers