# Initial condition for Kuramoto-Sivashinsky

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.

• There is a typo in your equation. – user33403 Jan 17 '20 at 21:06
• Indeed, thanks. – user33890 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? – Alone Programmer 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