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I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also need to predict the criteria when its inviscid(which basically makes it non-linear advection equation).

The FD schemes that I am using are 1st order FT for time, 1st order BS for the advection term and 2nd order CS for diffusion term. Though I know there are better FD schemes, I still want to analyze stability of this scheme.

Update
Forgot to include initial and boundary condition. Everything is so meaningless without it..
The domain of flow is between -2<=x<=8
Initial conditions are such that 1<=u<=2 for the entire flow domain
Boundary conditions define constant positive velocity at left and right boundaries.

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  • $\begingroup$ I'm not sure what you mean by FT, BS, and CS. I suppose these are low-order forward, backward, and centered differences? $\endgroup$
    – David Ketcheson
    Oct 10, 2013 at 5:49
  • $\begingroup$ Sorry about that..FT stands for forward in time...BS means backward in space....these are from upwind method and CS stands for central in space...I am trying to convey the FD approximations that I used for each derivative.. $\endgroup$
    – tumchaaditya
    Oct 10, 2013 at 7:06
  • $\begingroup$ A good reference is Randy LeVeque's book on hyperbolic equations $\endgroup$
    – Guido Kanschat
    Oct 10, 2013 at 19:52

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A useful rule of thumb (though it is not always a sufficient condition for stability) is stability of the linearized scheme. Since you have a method of lines discretization, you can think of this geometrically as the condition that the eigenvalues of the jacobian of your spatial discretization, multiplied by the time step size, lie inside the region of absolute stability of your time discretization. In your case, this will give you the same condition that you would have for stability of an advection-diffusion problem, but with the advection speed given by the largest value of the initial data.

Of course, your backward difference in space will only be stable if the initial data is non-negative.

For sufficient conditions for stability of discretizations of nonlinear PDEs, see Strang's paper.

In the inviscid case, it's well known (and follows from the reasoning above) that this scheme is stable for CFL number $\le 1$.

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  • $\begingroup$ Indeed, looking at linearized advection and diffusion CFL constraints and applying a safety factor is what people do for N--S as well. $\endgroup$
    – Rhys Ulerich
    Oct 10, 2013 at 20:33

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