What are good references for time-stepping of the coupled incompressible Navier-Stokes-heat equation (Boussinesq flow), $$ \begin{cases} \rho\left(\dot{\mathbf{u}} + \mathbf{u}\cdot\nabla \mathbf{u}\right) = -\nabla p + \mu\Delta \mathbf{u} + \rho(\theta) \mathbf{g},\\ \nabla\cdot \mathbf{u} = 0, \end{cases}\\ \begin{cases} \rho c_p \left(\dot{\theta} + \mathbf{u}\cdot\nabla\theta\right) = \nabla(\kappa\nabla\theta) + f \end{cases} $$ While the solution of either of the equations is well elaborated upon, and (the pitfalls of) the spatial discretizations are understood by and large, articles on time-stepping are harder to find. It is not clear to me, for example, when it makes sense to treat the nonlinear terms implicitly.

This question may or may not be treated in a more general sense, e.g., $$ \dot{u} = \mathcal{L}_1(u, v) + \mathcal{N}_1(u, v),\\ \dot{v} = \mathcal{L}_2(u, v) + \mathcal{N}_2(u, v). $$ (with linear terms $\mathcal{L}_*$ and nonlinear terms $\mathcal{N}_*$).

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    $\begingroup$ This may be one of the most covered topics in computational mechanics. There must be thousands of books and tens or hundreds of thousands of articles that cover the topic. Can you suggest particular aspects of the problem and solution methodology that are of interest to you that could narrow your question a little? $\endgroup$
    – Bill Barth
    Sep 17 '13 at 13:16
  • $\begingroup$ You may want to investigate Newton-Krylov-Schwarz methods. $\endgroup$
    – Paul
    Sep 17 '13 at 13:24

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