Numerical Analysis of a Higher Order Time Relaxation Model of Fluids

Abstract

We study the numerical errors in finite elements discretizations of a time relaxation model of fluid motion:
                          $u_t + u\cdot \nabla u + \nabla p - \nu\Delta u + \chi u^* = f$ and $\nabla \cdot u = 0$
In this model, introduced by Stolz, Adams, and Kleiser, $u^*$ is a generalized fluctuation and $\chi$ the time relaxation parameter. The goal of inclusion of the $\chi u^*$ is to drive unresolved fluctuations to zero exponentially. We study convergence of discretization of model to the model's solution as $h$, $\Delta t \rightarrow 0$. Next we complement this with an experimental study of the effect the time relaxation term (and a nonlinear extension of it) has on the large scales of a flow near a transitional point. We close by showing that the time relaxation term does not alter shock speeds in the inviscid, compressible case, giving analytical confirmation of a result of Stolz, Adams, and Kleiser.