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Tham khảo tài liệu 'a finite element scheme for shock capturing part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | fjt yt -inầtíừ eịi xK where the superscript n indicates the time-step and the subscript j is the spatial node location. We now present the results of this analysis for a 1 2 and for the temporal derivative parameter a of 1.0 and 1.5. We shall compare the relative amplitude and relative speed for a single time-step. The parameter for relative speed is given by tan 1 _y2jr Im L N _ 2ir relative speed 61 c N where N elements per wavelength _ _ . kAt c Courant number a __ Ax X. wave speed either Xy or X-2 For 1 which is first-order backward difference in time the relative amplitude is shown in Figure 29 and the relative wave speed is shown in Figure 30. This is plotted versus the number of elements per wavelength N and the Courant number c. Also remember that these comparisons apply for either characteristic Xy or X-2 even for subcritical conditions in which is negative. In these figures the Courant number varies from 0.5 to 2.0 and the elements per wavelength from 2 to 10. The amplitude portrait shows substantial damping for larger c and for the shorter wavelengths or alternatively the poorer resolution . The large damping at a wavelength of 2 Ax is important as this is the mechanism that provides the energy dissipation to capture shocks. Now consider the phase portrait or in this case the relative speed portrait. Over the conditions shown the numerical speed is less than the analytic speed throughout. For larger c the relative speed is somewhat lower worse . For N 2 the speed is 0 so that undamped oscillation could remain at steady state. 45 Chapter 3 Testing Figure 29. Relative amplitude versus c and resolution for at 1.0 and a 0.5 Figure 30. Relative speed versus c and resolution for ctf 1.0 and a 0.5 46 Chapter 3 Testing In comparison to the results we have shown in Figures 6-11 for Case 1 analytic shock case we must remember that Cs is the Courant number based on shock speed whereas c is based on the perturbation wave speed. If we consider a wave moving
