I am familiar with the Courant-Friedrich-Lewy Condition in as far as it applies to the stability of explicit finite difference schemes for standard parabolic and hyperbolic PDEs. However, when solving the Navier-Stokes Equations, the non-linearity of the PDEs make it difficult to derive a CFL condition analytically. I see that the "Courant number" is often estimated by the CFL condition of the 1st order wave equation or the residence time. Is it common practice to do so for all Reynolds numbers? Are there some Reynolds numbers under which an alternative formula is used to estimate the Courant number?
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4 Answers
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Strictly speaking, it doesn't make sense to speak of a Courant number for the Navier-Stokes equations at finite Reynolds number. The Courant number is defined in terms of a characteristic velocity, and solutions of parabolic equations (like Navier-Stokes) aren't described by characteristics. The CFL condition implies that no explicit solver with finite stencil width can be convergent (for parabolic problems like Navier-Stokes) when the ratio $\Delta t/\Delta x$ is fixed.
Nonlinearity doesn't cause any problem in defining the Courant number for hyperbolic PDEs, since you can still find the maximum characteristic velocity.
answered Jul 25, 2012 at 22:00
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$\begingroup$ Do you have a reference that you like which explains the "no explicit solver with finite stencil width" bit? $\endgroup$ Commented Jul 26, 2012 at 19:34
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2$\begingroup$ Yes. The original CFL paper! stat.uchicago.edu/~lekheng/courses/302/classics/… $\endgroup$ Commented Jul 27, 2012 at 9:48
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2$\begingroup$ @RhysUlerich See also people.maths.ox.ac.uk/trefethen/4all.pdf $\endgroup$ Commented Jul 27, 2012 at 19:38
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For the compressible Navier-Stokes equations, a convenient possibility is to choose the time step by considering the stability criterion of the inviscid (hyperbolic, Euler) and the viscous (parabolic, diffusive) terms independently from one another:
For the Euler term you have "something like"
$ \Delta t_\mathrm{Hyperbolic} \leq \displaystyle\min_{\mathbf{x}_i\in\Omega} \left( \min_{d=1,2,3} \frac{C \Delta x_{i,d}}{\lvert u_{i,d} \rvert + a_i} \right ) \; ,$
depending on your grid, spatial discretization and temporal discretization schemes, where $u_{i,d}$ is the d-th velocity component in the i-th cell/element, $a_i$ is the speed of sound, $\Delta x_i$ is the i-th cell/element size, and $C$ is the Courant number. Remember that in 1D the eigenvalues of the flux Jacobian for the inviscid term are $u + a$, $u - a$, and $a$ (and that you can compute them at every time step for every cell/element).
For the diffusive term you will have "something like"
$ \Delta t_\mathrm{Parabolic} \leq \displaystyle\min_{\mathbf{x}_i\in\Omega} \left( \min_{d=1,2,3} \frac{ \Delta x_{i,d}^2}{\lambda \; \mu} \right ) \; ,$
where $\mu$ is your diffusivity and $\lambda$ is a factor that depends on your spatial and time discretization schemes. Although it seems counterintuitive, explicit time integration schemes for the linear diffusion equation satisfying this criterion are stable for a sufficiently large $\lambda$. Implicit schemes allow you to take a larger $\Delta t_\mathrm{Parabolic}$ from the stability point-of-view, but if it is not small enough your scheme won't be very accurate.
So one could choose $\Delta t = \min ( \Delta t_{\mathrm{Hyperbolic}}, \Delta t_{\mathrm{Parabolic}} )$ and hope to remain stable for a sufficiently large class of problems if $C$ and $\lambda$ are choosen appropiately (they depend on the spatial and temporal discretization schemes!).
However, don't forget that: both terms are actually coupled, and as @Jed Brown said above, the CFL condition is a necessary condition (which is not the same as sufficient!). If you are using a first order finite volume method, a consistent and monotone numerical flux ensures that your scheme is both stable and convergent.
We know from Godunov theorem that no linear monotone scheme of higher than first order accuracy exist. This means that a higher order scheme has to be non-linear in order to be monotone. Total Variation Diminishing (TVD) methods can be constructed through limiting in FV, limiting and filtering in DG, and non-linear stabilization in FE. For higher than second order accuracy, it is worth relaxing the TVD constraint in order to retain the global order of accuracy of the method when discontinuities are present in the solution. Alternatives are Total Variation Bounded (TVB) and Total Variation Bounded in the Mean (TVBM). These methods do not remove oscillations completely, but kind of ensure stability while maintaining your global order of accuracy.
So, if you satisfy both the CFL condition and some non-linear stability condition (TVD,TVB,TVBM) your scheme will be very likely to remain stable. If you do not, you are on your own, which does not mean that your scheme will be unstable (see for example second-order FV schemes for the Euler equations using unlimited weighted least-squares reconstruction).
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$\begingroup$ I would add that for multi-species NS equations, one really needs to consider all diffusivities for stability. In a flow with very light species such as H or H_2, the mass diffusion might be limiting on very fine grids. Viscosity is the one most often cited in this context, but there are situations in my experience where mass diffusion needs to be considered too. $\endgroup$ Commented Jul 28, 2012 at 6:09
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This estimate is not okay when the viscous terms become large, either at low Reynolds number or in (possibly very stretched) boundary layers. For linear stability, you can use
$$\min \Big[ C_1 \frac{\Delta x}{u}, C_2 \frac{(\Delta x)^2}{\eta} \Big]$$
where $u$ is velocity, $C_1$ depends on the spatial discretization of the transport terms (usually $\frac 1 2$ or $1$ for the simplest spatial and temporal discretizations), $\eta$ is the viscosity, and $C_2 = \frac 1 2$ for the simplest spatial and temporal discretizations. Note that CFL is only a necessary condition for linear stability, it may not be sufficient. Preservation of nonlinear stability may impose more stringent time step constraints, e.g. in the case of strong stability preserving methods.
answered Jul 25, 2012 at 20:24
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$\begingroup$ I'm not sure I understand the difference between linear and non-linear stability... Are you referring to whether the equations are linear/non-linear? $\endgroup$– PaulCommented Jul 25, 2012 at 20:44
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$\begingroup$ But the viscous terms are usually solved either (semi-)implicitly, or by a more stable multistep method. $\endgroup$ Commented Jul 25, 2012 at 21:19
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$\begingroup$ @VladimirF Of course that depends on the particular instance of the method. There are good reasons for both choices. Note that most splitting methods for Navier-Stokes are very inaccurate at low Reynolds number. Proving stability for semi-implicit (operator splitting) methods is much more involved. $\endgroup$ Commented Jul 25, 2012 at 21:29
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$\begingroup$ What other conditions can I impose to guarantee non-linear stability? Are there any heuristics on this? $\endgroup$– PaulCommented Jul 27, 2012 at 17:00
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$\begingroup$ @Paul You have to define what you want. $\endgroup$ Commented Jul 27, 2012 at 19:39
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If you're looking for a time step stability criterion in the nondimensional compressible Navier--Stokes setting, Guarini derives an estimate of it using scalar stability theory in section 4.2.2 of his thesis.
Check out Guarini, Stephen. Direct numerical simulation of supersonic turbulent boundary layers. PhD thesis, Stanford University, 1998. http://proquest.umi.com/pqdweb?did=732826501&Fmt=7&clientId=48776&RQT=309&VName=PQD.
