Numerical Analysis of a Higher Order Time Relaxation Model of Fluids

We study the numerical errors in finite element discretizations of a time relaxation model of fluid motion: ut + u · ∇u + ∇p − ν∆u + χu∗ = f and ∇ · u = 0 In this model, introduced by Stolz, Adams and Kleiser, u∗ is a generalized fluctuation and χ the time relaxation parameter. The goal of inclusion of the χu∗ is to drive unresolved fluctuations to zero exponentially. We study convergence of discretization of the model to the model’s solution as h, ∆t → 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.