I have a homework problem that asks me to show that 1st order unwinding or central differencing can give a strongly conservative, consistent scheme for the 1-D Burger's Equation using a finite volume approach. I think I understand unwinding schemes and central differencing, but I'm not sure what "strongly conservative means or how to demonstrate it.

  • $\begingroup$ I don't actually know what "strongly conservative" means. But conservative is in my opinion that for the sum over flux terms only boundary terms remain. Hence $$\sum_{i=1}^{N}f_i=f_N-f_1\,.$$ This is also called a telescopic sum. For the pde you find $$ \frac{\partial}{\partial t}\sum_{i=1}^{N}u_i+f_N-f_1=0\,.$$ $\endgroup$ – sebastian_g Feb 1 '15 at 10:46
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    $\begingroup$ For reference, the correct term is "upwinding", not "unwinding". $\endgroup$ – Wolfgang Bangerth Feb 1 '15 at 17:13

Strong conservative in 1D usually means that the change in the solution is equal to the flux in minus flux out. For example, for Burgers' equation without source term, you can rewrite it as

$u_{,t} + (u^2)_{,x}/2 = 0$

If you integrate both sides in $x$ over an interval $[a,b]$, you have

$\int_a^b u_{,t} = u^2(a)/2 - u^2(b)/2$.

In other words, $u^2/2$ is the flux, and the total rate of change in $u$ over an interval $[a,b]$ is equal to the flux in at one end minus flux out at the other.

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