ADER-WENO schemes and the 2D HLL Riemann solver. Figure 7 shows the same Alfven wave propagation test as Figure 6 but when a lower quality Riemann solver is used. We used the same ADER-WENO schemes as in Figure 6. The only difference is that the simulations in this Figure were run with a 1D HLL Riemann solver and a 2D HLL Riemann solver. Comparing Figure 6 to Figure 7 enables us to appreciate the improvement that Riemann solvers with sub-structure provide in reducing dissipation.