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I am trying to solve the Navier Stokes equations using the finite element method. I plan on using the pressure correction method to deal with the pressure and an implicit time stepping scheme for diffusion and an explicit time stepping scheme for convection. My question is: How does one deal with the non-linear term using the finite element method? I have investigated a couple approaches. For example, I understand for steady problems it is common to use a Newton iteration. Since I am interested in unsteady problems I don't think this method is suitable. I have also seen an approach where part of the non-linear term is "frozen" at the last time step so that at each time step the problem appears as a linear advection problem, i.e.

$u_{i}\frac{\partial{u_{i}}}{\partial{x_{j}}} \approx{} u_{i}^{n-1}\frac{\partial{u_{i}^{n}}}{\partial{x_{j}}}$

In other words the velocity at the previous time step ($u_{i}^{n-1}$) is treated as constant and no shape function expansion is used in approximating this term. Is this method any good? What methods should I use to deal with the non-linear term in the unsteady Navier Stokes equations where accuracy, ease of implementation, stability, and robustness to different Reynolds numbers are all considerations?

Thanks

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You can absolutely use the Newton method to linearize the system of equations that results after you have discretized in time. It's a pretty common approach. It might be overkill in lots of cases, but it's not inappropriate. Freezing like you suggest and not iterating at all within a timestep can lead to some pretty awful solutions unless your timestep is miniscule. But you don't want to do that since you are interested in implicit approaches.

Write the split equations, and then discretize in time to give a non-linear PDE in space. Then apply Newton's method. Then discretize in space with FEM.

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  • $\begingroup$ I knew the Newton method could be use for the unsteady case, but was under the impression that it was not that common because it was too costly to perform a Newton iteration at every time step. I guess I was wrong. Do you happen to have a good reference/paper that works through this problem for the unsteady case? Thanks for your help. $\endgroup$ – James May 20 '15 at 1:49
  • $\begingroup$ Unfortunately, I was never a pressure-projection method proponent, so I don't have one in that context. I just used Taylor-Hood elements (LBB stable) and a good preconditioner that could deal with the indeterminate linear system. $\endgroup$ – Bill Barth May 20 '15 at 1:51
  • $\begingroup$ And of course, I meant "indefinite". $\endgroup$ – Bill Barth May 20 '15 at 2:11
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Addendum to Bill's answer: a Newton iteration combined with implicit timestepping usually regularizes the problem enough to allow just one Newton iteration per timestep (for one example - http://users.ices.utexas.edu/~benkirk/dissertation.pdf). With a decent preconditioner, this can be competitive with pressure projection if the timescale of your physics isn't too small.

To your question: if you're doing unsteady flows, the nonlinear term is usually dealt with in an explicit manner, as it's roughly equivalent to a convective term. So long as you treat the nonlinear properly as a linear convection equation with non-constant convection (i.e. aliasing, higher order quadratures, etc), this is usually fine.

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  • $\begingroup$ I was planning on using an explicit time stepping scheme for the convection and only using implicit for the diffusion. When you say "...treat the nonlinear properly as a linear convection equation with non-constant convection..." do you mean that $u_{i}\frac{\partial{u_{i}}}{\partial{x_{j}}}\approx{}u_{i}^{n}\frac{\partial{u_{i}^{n}}}{\partial{x_{j}}}$ where $u_{i}^{n}$ is treated as being a constant at each node (and thus only the $u_{i}^{n}$ in $\frac{\partial{u_{i}^{n}}}{\partial{x_{j}}}$ is expanded with shape functions? $\endgroup$ – James May 20 '15 at 4:05
  • $\begingroup$ The nodes really refer to points where piecewise polynomial hat functions are joined together. If you're using just piecewise linear (standard) FEM, then yes - expanding $u^n_i$ using shape functions at each node should be OK since $(u^n_i)_{,x_j}$ is constant. $\endgroup$ – Jesse Chan May 21 '15 at 13:10
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You say you use an explicit time stepping scheme for convection. Then you don't need a non-linear solver at all.

$$ \frac{u_i^{n+1}-u_i^n}{\Delta t} + u_j^n \frac{\partial u_i^n}{\partial x_j} + \frac{\partial \tilde{p}}{\partial x_i}- \nu \nabla^2 u_i^{n+1} = 0 $$

where subscripts denote spatial coordinate directions (i = 1,2,3 in 3D), and superscripts, the time step. This scheme is consistent, and is called a semi-implicit scheme. The non-linear term can be transferred to the right hand side and just becomes part of the residual at each time step.

If you want a fully implicit scheme without Newton iterations, you could use the "characteristics-Galerkin" method. This is a semi-Lagrangian method in which you approximate the total derivative implicitly: $$ \frac{Du}{Dt}|^{n+1} \approx \frac{u^{n+1} - u^n \circ X(x,t^n)}{\Delta t} $$ where $X$ denotes the characteristic passing through $(x,t^{n+1})$, which can be computed, for instance, by solving an ODE. (So finally this may not be much cheaper than a Newton method.) For details, you can check out

Mofdi El-Amrani and Mohammed Seaid, "An L2-projection for Galerkin-characteristic solution of incompressible flows", SIAM J. Sci. Comput., Vol 33 No 6, pp 3110-3131, 2011.

You can try out the method in FreeFEM++, if you want to. It has a convenient "convect" operator for this.

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