Higher order vs First order numerical schemes for conservation laws

Question

I am trying to understand higher order methods for scalar conservation laws given by $u_t+f(u)_x=0$

In which sense these higher order schemes perform better than first order schemes?

For example Lax Wendroff scheme though it is second order when I compare the results of this scheme with that of Godunov Scheme which is of first order, Godunov scheme's solutions look much better than that of Lax Wendroff on computer (Lax Wendroff scheme has oscillations where as Godunov scheme does not) ..

So I am not understanding in which sense they are better and what are there practical use?

What are the properties that a higher order scheme possesses that is not present in first order schemes like Godunov..

Answer 1

Second-order schemes have a higher-order global truncation error. That is to say, if the true solution is smooth, then the numerical solution converges faster to the analytical solution as the mesh size is decreased. However, order of accuracy and convergence speed are not everything. Indeed, there are other artifacts such as dispersion and attenuation, which may cause oscillations. Thus, despite Lax-Wendroff converges faster than Godunov towards smooth solutions, it doesn't work as well as Godunov's method on non-smooth solutions. Now, we can actually combine the two schemes to take advantage of both methods (see literature on high-resolution methods, such as flux limiter and slope limiter methods).