[Panee]

Hey all!

I am trying to solve the 1D Burgers equation using finite differences.

1. Inviscid - I have tried to solve it using Upwind, Maccormack and Lax Friedrichs scheme. Maccormack gives heavy oscillations near the shock region and Lax Friedrichs gives a damped solution. Although Upwind seems to work for Riemann conditions, when I give a sinusoidal initial condition, the method blows up (I dont think the method can handle both positive and negative values at the same time). So what do I do to create a solver which handles the discontinuities as well as smooth initial conditions without any damping?

2. Viscous - Is Cole Hopf transformations valid only for f(0,t) = g(L,t) = 0 at the boundaries? If not, what changes should I make to the transformation?

Thanks in advance!

[lex78700]

Hi Panee,

I am not sure to understand your question 2, so I will focus on 1.

To my knowledge, no finite difference scheme can handle discontinuities without damping. Although the inviscid Burgers equation seems simple, it is actually very hard to solve for an sinusoidal initial condition. Physically, Burgers says that the fluid accelerates where the velocity is larger. In a sine wave, you have positive velocity behind an area where the velocity is negative. It means that the fluid behind will accelerated whereas the fluid in front of it will slow down. This will obviously lead to convergence of the fluid and create a shock. Analytically, the solution after a given time (the time needed for the shock to form) doesn't exist anymore (in the usual mathematical sense, that is a continuous solution). So the math says your solution blows up, and you should expect a good numerical code to give you this result (the real solution) as well.

The oscillations or damping you see in your codes are due to numerical errors associated with your finite difference schemes. These schemes only approximate the real analytical solution, and due to their finite order (hence the name finite difference), they introduce errors. Some schemes are called dispersive (they introduce spurious waves) while the others are called dissipative (they damp the solution and especially gradients). This is normal to see such errors, and you can reduce them by choosing higher-order schemes, or schemes that are best adapted to your specific problem, or by reducing the time step or increasing the spatial resolution.

But this won't solve your problem. After a given time, an initial sine wave will eventually blow up no matter which numerical method you use. You could delay the blow-up by using a super-diffusive scheme (since the high gradient preceding the blow up will make a dispersive finite difference scheme crazy, cause they hate that), but this would spoil you whole solution and I guess you wouldn't be happy if you end up with a zero solution after 1 second (that's what overdamping would cause).

Of course many people have to simulate shock waves (the kind of discontinuity waves generated by Burgers equation that you observe) but they don't use explicit finite differences for that (as I explained, it's not suited). They use special schemes based on the finite volume method which are specifically designed to handle and propagate such waves and conserve some flow properties that are of interest (for example vorticity). I'm not sure that's what you're looking for, but this is a whole area of research (it does exist!).

I hope I was clear, don't hesitate if you need more help!

Adrien

[FMDenaro]

Many details about the Burgers equations can be found in the book of Leveque about FV for hyperbolic equations.
Note that the original paper of Burgers illustrated an equation with the diffusive term. For small viscosity, you can use a finite volume scheme based on upwinded flux, using a very small mesh size (Reh=O(1))

[Panee]

Thanks a lot Adrien and FMDenaro!

Adrien, surprisingly, Lax Friedrichs scheme is producing a bounded solution with progressing time, although the shock is not as sharp as I would like.

FMDenaro, I will go through the book by LeVeque, thanks!

[shahid_qa32]

Hy to everyone,

I am using Crank Nicolson for 1D Burger.I am getting confuse to use Initial guess as Initial condition?

Answer will be highly appreciated