Implicit/Explicit schemes in Stabilization methods

Stabilization methods such as Streamline upwind Petrov–Galerkin and Galerkin-Least-Squares typically contain a stabilization parameter that looks something like

When applying this method to the Navier-Stokes equation, what is the treatment for the velocity term \mathbf{u} within \tau_{M,T}? It is a term that can bring convergence issues I think. Is it better to treat this term explicitly in a nonlinear method such as Newton? Or is it better to include it when calculating the jacobian of the entire variational form?

My anecdotal experience with this is that it’s fine to use the full Jacobian of the SUPG terms in Newton’s method, although many manual implementations simply treat \tau as a constant when computing the Jacobian, which involves slightly fewer floating-point operations during assembly and is easier to implement when you don’t have access to FEniCS’s derivative function. However, using the exact Jacobians of highly-nonlinear shock-capturing viscosities is likely to lead to convergence issues.

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