# What does “strongly conservative” mean in the context of numerical methods?

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.

• 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 $$\frac{\partial}{\partial t}\sum_{i=1}^{N}u_i+f_N-f_1=0\,.$$ – sebastian_g Feb 1 '15 at 10:46
• For reference, the correct term is "upwinding", not "unwinding". – Wolfgang Bangerth Feb 1 '15 at 17:13

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