I am not sure if this is the correct place to post my question, so please correct me if there is a better site. I recently started applying some discretization schemes such as Upwind, Lax-Wendroff with 1 and 2 steps, and surprisingly, one can see that these schemes may have a dispersive or diffusive character which is 100% related to the numerics and not to the physics of the system.
Can any one explain how/why does this happen? How come a discretized PDE may be completely different then the original continuous PDE?
Regards.
I am supposing you are trying to get your hands on some numerical methods for hyperbolic conservation laws, for example, the conservation of mass in the Euler equations: $$ u_t + \mathrm{div}\mathbf{F}(u) = 0, $$ where $\mathbf{F}(u)$ is the flux, for example $\mathbf{b}\cdot \nabla u$. To make the problem more simple, we can consider a square wave $u$ traveling to the right satisfying $$ u_t + au_x = 0. $$
Upwind: choosing upwind as fluxes results the numerical solution getting smoothened out over time marching, because this scheme is not flux conservative, meaning the local conservation of the flux is not satisfied: the numerical solution $u_h$ does not satisfy $$ \int_{V} \big(u_t + \mathrm{div}\mathbf{F}(u)\big) dx = \int_{V} u_t\, dx + \int_{\partial V} \mathbf{F}(u)\cdot \mathbf{n} dS = 0, $$ which physically means that the mass flowing into any small region $V$ at any time $t$ should equal to the change of mass within this region. The "smoothened out" effect is sometimes called dissipation, or diffusive in your term.
Lax-Wendroff: now you have local conservation, but the boundary of the square wave solution will get "wavy", oscillatory behavior as marching in time. This is dispersion.
Dissipation or dispersion happens a lot in the computation of Euler equations. The reason is that the discretized PDE doesn't preserve the some key features of the continuous problem.