Example A.2: Burgers' Equation

In this example we will study Burgers' equation

$$u_t + (u^2/2)_x/ = 0$$

which is the archetypical example of a nonlinear equation, possessing a convex flux that may cause discontinuous shock waves to form even for smooth initial data. We assume Neumann boundary conditions and use a single shock as initial data.

For this particular setup, we know the exact solution

T  = 1.2; NN = 512; hh=1/NN; xx=hh*(1:NN);
uexact = 1.0*(xx<(0.1+0.5*T));

We then solve the equation numerically using two first-order methods (Lax-Friedrichs and the upwind method) and two second-order methods (Lax-Wendroff and MacCormack)

N  = 32; h=1.2/N; x=h*(1:N);
u0 = 1.0*(x<0.1);

method = {'LxF', 'upwind', 'LxW', 'McC'};
name   = {'Lax-Friedrichs', 'Upwind', 'Lax-Wendroff', 'MacCormack'};
for i=1:4
   plot(xx,uexact,'-',x,u,'o','MarkerSize',4); axis([0 1 -0.2 1.3]);

Comparing the two first-order schemes, we see that the upwind scheme resolves the discontinuity quite sharply, whereas the Lax–Friedrichs smears it out over several grid cells. Both second-order schemes resolve the discontinuity sharply, but produce spurious oscillations upstream.