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For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation of which I know little. I just know that it was derived the equation to model the diffusive instabilities in a laminar flame front. It reads

$$ u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0. $$

Obviously I need an initial condition $u_0$ but I don't know what to choose. I would like to simulate something with a physical meaning.

I'm also looking for advice and documentation on this equation.

Thank you in advance.

  • There is a typo in your equation. –  Jan 17 '20 at 21:06
  • Indeed, thanks. –  Jan 17 '20 at 21:09
  • It looks a like a 1D Navier-Stokes equation plus a higher order term ($\eta u_{xxxx}$). Your computational domain is just a line and you should have an inlet located in the left side ($x = 0$) for example and an outlet located in the right side ($x = L$, where $L$ is the length of your line). So, it's reasonable if you assume that velocity is zero everywhere in your line when $t = 0$ and find where the flame front will start to become unstable. Does it sound like a reasonable scenario to you? – Mithridates the Great Jan 17 '20 at 22:00
  • If u = 0 at initial conditions, if you use a periodic bc (which is common for modified ks equations) or dirichlet then you get a trivial solution. – EMP Jan 17 '20 at 22:32
  • Always a good answer: When you look through the literature for other people who have solved this equation, what do they use? – Wolfgang Bangerth Jan 18 '20 at 16:21

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So I've done a modified version of this equation for a different goal, and we usually use a randomized starting profile to show that it converges to an approximate steady state regardless of the initial conditions. It really depends on the type of situation you want to simulate. I like the randomization of the initial conditions, because I think that for this equation (as well as the modified equation) if you choose your bcs wisely you can show ergodicity of the equations.

EMP
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  • Ergodicity of the equation? What does that means? I like the idea of a randomized starting profile, but what boundary conditions will you use with that? –  Jan 18 '20 at 13:16
  • Ergodicity means the independence of the long time behavior to the initial conditions. The way to check that, is to check that the partial derivative of spatially averaged u with respect to time is nonzero, otherwise, the average u will always be the average of the initial condition. You can check this formulation yourself for different boundary conditions, but I myself have used homogeneous dirichlet and neuman boundary conditions at both the inlet and the exit. I'm not positive on how the classical KS equation will behave with those compared to the modified KS equations which are chaotic. – EMP Jan 18 '20 at 17:27
  • You convinced me, thanks for the detailed anwser –  Jan 19 '20 at 19:10