The uniform velocity advection problem is solved with a number of algorithms and input parameters. The refinement ratio between succesive grid levels is 2 in all the computations presented below.
The Lax-Friedrichs scheme introduces a large amount of numerical diffusion. When using a single level of grids the initial square pulse is rapidly diffused.
When using smaller step sizes the numerical diffusion is even larger
The effect is reduced when using multiple grid levels, but still noticeable
Upwind also introduces significant numerical diffusion, but less than Lax-Friedrichs. One can notice that the error estimation procedure generates grids that conform more closely to the initial square discontinuity than in the Lax-Friedrichs computation, a reflection of the smaller values of numerical diffusion.
The second order Lax-Wendroff scheme has a dispersive leading order error term. Numerical diffusion is less than in the previous schemes but the dispersive error leads to over/undershoots in the solution. The dispersive error subsides quickly with increasing resolution, a reflection of the scheme's second order of accuracy.
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