Implicit/Explicit schemes in Stabilization methods

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

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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?

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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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