# For a hyperbolic PDE, is there any proof that the BDF2 method is stable for integrating them?

I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs.

Say I have a hyperbolic equation as $$\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$$. This system is coupled to the Navier-Stokes equation (the governing equations for $$\bf U$$) : $$\frac{\partial {\bf U}}{\partial t} + {\bf U} \cdot {\bf \nabla}{\bf U}= -\nabla p + \frac{1}{Re}\nabla^2{\bf U} + c^2 {\bf j}$$, where $$\bf j$$ is in the vertical direction. I have applied BDF2 to these equations and the solutions seem stable.

The thing is that some people add an artificial diffusion term in the $$c$$ equation in order to secure the numerical stability (with some explicit time-integration method). My experience is that the BDF2 method is stable. Is there any proof that BDF2 is always stable in this case?

Thank you very much. Any help is appreciated.

• BDF2 has unconditional stability (for negative eigenvalues), so it should be fine. What's usually the bigger factor is the stability of the alpha-stable higher order BDF methods, since when you're being efficient you normally aren't just using BDF2. In that case, you'd look at the eigenvalues of the operator to assess the stability of the given method, whether it's high in the complex region is an indicator of instability for BDF methods. May 31 at 12:34
• For a purely hyperbolic PDE you may be in a bit of trouble here as BDF2 contains the imaginary axis in its boundary and the eigenvalues of advection equation will lie directly on the imaginary axis. It gets worse for other BDF methods which do not contain the imaginary axis. Your question however is not really about a hyperbolic equation as it is coupled to a parabolic equation. This is a very different situation that is much more difficult to say anything definitive about. If you would like to specifically know about hyperbolic PDE stability I would remove the NS equation part. Jun 1 at 1:06
• Are you asking about discretizing in time while keeping things continuous in space? Or (more likely) are you also discretizing in space? If so, the answer depends completely on your spatial discretization. Also, the need for artificial diffusion depends partly on your definition of "stable". Can you be more precise? Jun 1 at 5:32