The initial data for this www.selleckchem.com/products/Abiraterone.html problem are the sin2x??sin?y)(26)with?sin2x??sinz,?0),B=(?sin?ysinz,?sinx??sinz,?following:��=��2,p=��,u=(?sin?ysinz, 0 �� x, y, z �� 2��, where �� = 5/3. Again, grid independence is demonstrated in Figure 11 through density contours on slices across the centerlines planes on coarse (1283) and fine grids (2563). The results presented here in this section are those computed on the 2563 grid using the CWENO reconstruction without diagonal smoothing.Figure 11Instantaneous contours of density across x-, y-, and z-direction centerlines ( = ��) at t = 0.2 for two different grid sizes 1283 (a) and 2563 (b).A way of demonstrating the accuracy of a numerical method is to determine whether the solenoidal constraint ?B = 0 is maintained throughout the simulation.

Since ?B = 0 initially, theoretically it should remain so throughout the simulation. However, the accumulation of numerical errors can usually lead to nonphysical phenomenon known as magnetic monopoles (when ?B is not equal to 0). The schemes presented here when first introduced in [21] in 1D and 2D frameworks did not require an explicit enforcement of the solenoidal constraint for producing stable and reasonably accurate solutions, and hence no such treatment is used here either. Figure 12 shows the surface plots of ?B on all the Z-surfaces at a certain instant in time. The maximum value of the magnitude of ?B at this instant is 0.41, which is actually representative of the entire simulation.

Although these simulations did not show any kind of instabilities, the divergence values are reasonably large for such computations and go to show that in 3D some divergence treatment [32, 33] might be necessary. Figure 13 shows a density isosurface, and Figure 14 shows contours of density on three slices across the x = y = z = �� planes. Figure 15 shows the 2D magnetic field vector colored by magnetic field magnitude. These results are similar to those of previous studies [32, 33] and demonstrate the ability of such higher-order central schemes to resolve the shocks that the vortex system develops while maintaining the simplicity and ease of implementation typical of this black-box type of finite difference schemes.Figure 12Instantaneous surface plot of the divergence of the magnetic field (?B) on all the Z-surfaces at t = 0.5 for GSK-3 the solution of the Orszag-Tang system.Figure 13Isosurface of the density at a value ��C = 3.0 at t = 0.2.Figure 14Instantaneous contours of density across x-, y-, and z- direction centerlines ( = ��) at (a) t = 0.2, (b) t = 0.4, and (c) t = 0.8 (grid: 2563).