Aside from extra computational cost due to having to compute both fluxes over a certain region, is there any downside to blend two flux evaluations for a hybrid scheme in a finite volume method? The flux evaluation would look like this:

$\mathbf{F}_{i+\frac12} = \Lambda_{i+\frac12} \mathbf{F}^c_{i+\frac12} + (1 - \Lambda_{i+\frac12}) \mathbf{F}^u_{i+\frac12}$

The switch is based on a pressure and/or density gradient sensor depending on your application. $\mathbf{F}^c$ is a central scheme (McCormack, compact, ...) and $\mathbf{F}^u$ is an upwind scheme like a flux-difference splitting with a MUSCL reconstruction. Are there any issues in terms of numerics, conservative properties if I am blending the two schemes using a continuous function for $\Lambda$ as opposed to simply switching between schemes with $\Lambda$ valued as either 0 or 1?

  • $\begingroup$ The classic example of switches-gone-wrong is the 1970 Murman-Cole scheme for transonic potential equations (fixed in 1973). If your switch, or blending, doesn't yield a telescoping sum then you violate conservation. It's straight forward to test. $\endgroup$
    – tpg2114
    Nov 27 '12 at 19:15
  • $\begingroup$ How will you be solving the equations? Implicit (pseudo-)time discretization? Switching could then have a negative effect on the iterative convergence. $\endgroup$
    – chris
    Nov 29 '12 at 20:06
  • $\begingroup$ That would be using an explicit time scheme, either predictor-corrector or some basic Runge-Kutta. $\endgroup$ Nov 29 '12 at 20:08

The approach you are using will maintain conservation either way. There are other obvious approaches that are not conservative, and can cause problems.

It is possible (and even likely) that you will lose an order of accuracy in the region where you switch, if you examine the local truncation error. But typically this error is localized so that the global error is still of the expected order. So in my experience, you will see essentially the same behavior whether you use a hard switch or a transition region.

I have a manuscript on (more or less) this topic: Error Analysis of Explicit Partitioned Runge-Kutta Schemes for Conservation Laws .

I would be very interested in knowing what you see when trying the two approaches, if it is different from what I'm suggesting.

  • $\begingroup$ awesome, I was hoping you'd be able to comment on this. $\endgroup$ Nov 28 '12 at 21:56
  • $\begingroup$ Is there a theoretical way to check, akin to telescoping sums? $\endgroup$
    – tpg2114
    Nov 29 '12 at 0:13

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