CookBook%2FBurgersEquation

The following examples were originally provided to us by fred2 <at></at> qnet.com in http://thread.gmane.org/gmane.comp.python.fipy/911

The linearized Burgers equation, $$\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} - \mu \frac{\partial^2 u}{\partial x^2} = 0,$$ is a simple model equation for gas dynamics (Anderson, Tannehill, Pletcher [Anderson:1984]_, chapter 4). If $\mu = 0$, the wave equation is recovered. For the boundary conditions $$u(0,t) = u_0$$ $$u(L,t) = 0,$$ the exact steady-state solution is $$u = u_0 \left ( \frac{1-\exp{[R_L(x/L]}}{1-\exp{(-R_L)}} \right ),$$ where $R_L = cL/\mu$.

The following script integrates the equation (with $c=1/2$, $\mu=1/4$, and $L=1$) to large time.

Image(burg_linfig1.png)

For periodic boundary conditions, and the initial condition $$u(x,0) = \sin{kx}$$ the exact unsteady solution is $$u(x,t) = \exp{(-k^2\mu t)} \sin{[k(x-ct)]}.$$ The following script integrates the equation (with $c=1/2$, $\mu=1/4$, and $k=2\pi$) to moderate time on the interval $x\in [0,1]$.

Image(burg_linfig2.png)

The following script gives a solution to the steady-state, non-linear Burgers equation.

Image(burg_nonfig1.png)

Benton and Platzman [Benton:1972]_ describe 35 different exact solutions to the Burgers equation.