For the linear advection $u_t + au_x = 0$, we have the explicit Lax-Friedrichs scheme
$$ u_k^{n+1} = \frac{1}{2} (u_{k+1}^n + u_{k-1}^n) – a\frac{\Delta t }{2 \Delta x } (u_{k+1}^n – u_{k-1}^n) $$
But, if we replace this equation with $u_t + ( f(u) )_x = 0 $, the L-F method now reads
$$ u_k^{n+1} = \frac{1}{2} (u_{k+1}^n + u_{k-1}^n) – \frac{\Delta t }{2 \Delta x } (f( u_{k+1}^n) – f(u_{k-1}^n)) \tag{A}$$
If our $u_0(x)$ is smooth or piecewise smooth, then my ${\bf understanding}$ is that the method above will work just fine ${\bf unless}$ a shock or rarefaction forms in which case the exact solution is not longer a classical but a weak solution and the numerical method may not converge. But, I have learnt that if we can put our method in the form
$$ u_k^{n+1} = u_k^n – \frac{ \Delta t }{\Delta x} [ F(u_k^n, u_{k+1}^n) – F(u_{k-1}^n, u_k^n)] \tag{B}$$
and $F(u,w)$ is called ${\bf numerical \; flux}$, then we guarantee our method wont converge to a non-solution. Now, my books claims that L-F can also be written in conservative form if we take
$$ F(u_k, u_{k+1}) = \frac{ \Delta x }{2 \Delta t} ( u_k – u_{k+1}) + \frac{1}{2} ( f(u_k) + f(u_{k+1}) ) $$
and I assume this is done by some manipulation of equation (A). But, here is where my confusion arises. Isnt equation (B) and (A) just the same? What is special about equation (B)? Can someone clarifies this to me?
Best Answer
Both conservative and non-conservative Lax-Friedrichs schemes are identical. To see this, one injects the expression of the numerical flux in $(\text B)$, which gives $(\text A)$.
When considering smooth solutions, one can either use $u_t + f'(u) u_x = 0$ or $u_t + f(u)_x = 0$ to develop numerical methods that will converge towards the strong solution. However, in case of nonsmooth solutions it is advantageous to use only the conservative PDE $u_t + f(u)_x = 0$ to develop numerical methods, more precisely its integral form $$ \frac{\text d}{\text d t} \int_{x_1}^{x_2} u\,\text d x = f (u|_{x=x_1})- f (u|_{x=x_2}) , $$ which leads to the definition of conservative methods. According to the Lax-Wendroff theorem, a stable conservative method will converge towards a weak solution of the PDE. Hence, discontinuities will satisfy the Rankine-Hugoniot condition, i.e., shock waves will propagate at the correct speed. Note that this is not necessarily true for non-conservative methods (see e.g. ยง12.9 pages 237-238 of (1)).
(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge university press, 2002. doi:10.1017/CBO9780511791253