Can we use the von Neumann stability analysis to investigate the stability of the discrete form of the following problem?
$$u_t+\frac{\partial(x^2u)}{\partial x}=S(u,x)\ .$$
Please give some hint how to use this analysis in the above problem.
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$\begingroup$ Is the function $S(u, x)$ linear on $u$? $\endgroup$– nicoguaro ♦Commented Jul 27, 2017 at 1:08
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$\begingroup$ @nicoguaro yes $S(u,x)$ is linear on u $\endgroup$– SandyCommented Jul 27, 2017 at 1:17
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$\begingroup$ You can follow this example in Wikipedia. $\endgroup$– nicoguaro ♦Commented Jul 27, 2017 at 1:35
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$\begingroup$ @nicoguaro I know very well when a is constant. But my question is when flux function is of the above type $\endgroup$– SandyCommented Jul 27, 2017 at 4:35
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$\begingroup$ @Sandy you'll want to look at the so-called method of "frozen coefficients". this older scicomp post is a good starting point $\endgroup$– GoHokiesCommented Jul 27, 2017 at 12:09
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2 Answers
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Im not sure there is a rigorous justification, but consider that stability is defined (should be defined) in the following way. That the numerical solution remains bounded as the mesh sizes in time and space tend to zero. So stability is about what happens on fine meshes. On fine meshes, nonlinear problems such as the one you offer "look more linear".
There are other methods of defining stability in the nonlinear case such as energy stability. As far as I know, the outcome is always consistent with von Neumann stability.
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$\begingroup$ How is this problem nonlinear? OP said the forcing term is linear in $u$. $\endgroup$– GoHokiesCommented Jul 30, 2017 at 14:22
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$\begingroup$ @GoHokies What is the question? Flux is nonlinear $\endgroup$ Commented Jul 30, 2017 at 20:39
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$\begingroup$ @PhilipRoe no. the flux function is linear in $u$. $\endgroup$– GoHokiesCommented Jul 31, 2017 at 6:17
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$\begingroup$ @PhilipRoe ... hence this is just a linear advection PDE with a non-constant advection coefficient ($u_t + a(x)u_x = \bar{S}(u,t)$ with $a := x^2$ and linear $\bar{S} := S - 2xu$) . $\endgroup$– GoHokiesCommented Jul 31, 2017 at 6:29
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1$\begingroup$ @GoHokies The non-constant advection is precisely what makes it nonlinear. What you call the method of frozen coefficients is what I would call a local linearization. AFIK, there is no rigorous mathematical justification for this, but it work with amazing reliability. The reason for this is that the CFL is not a mathematical condition but a physical condition, stating that the numerical domain of dependence must include the analytical domain of dependence. This is a local condition. It is clearly necessary but maybe not sufficient. $\endgroup$ Commented Jul 31, 2017 at 8:54
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Use the chain rule to re-write your original PDE as a inhomogeneous advection equation with variable advection speed:
$$u_t + x^2 u_x = S(u,x) - 2 x u.$$
Then use the method of frozen coefficients to investigate stability of the corresponding discrete PDE problem (taking into account, of course, any boundary and initial conditions you may have). Check out this older scicomp post. Here's a couple more references to get you started:
http://www.physics.arizona.edu/~restrepo/475B/Notes/sourcehtml/node37.html
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$\begingroup$ what will be the CFL condition in the above problem if we modify the above equation as u suggested. $\endgroup$– SandyCommented Jul 30, 2017 at 10:18
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$\begingroup$ @Sandy you ignore the RHS (as that is irrelevant when considering discrete stability), and apply the method of "frozen coefficients" for the discrete left-hand side, as shown on pages 3-4 of the slides i linked to. $\endgroup$– GoHokiesCommented Jul 30, 2017 at 14:31