Rectangular pulse = 1 magic time-step S Fig. 1. Position of the rectangular pulse at fixed time intervals for 1 (magic time-step). = 160 em_swe_03.m S = Nt Figure 1 shows the position of the pulse after fixed time intervals when the time step is set equal to the magic time step where the Courant number is = 1. The shape and spatial width of the pulse are completely S preserved as it propagates along the X axis. The free space velocity is a.u. The numerical velocity can be measured from the displacement of vn the pulse from = 0 a.u. and at the end of the execution time = 8.00 dx t dt a.u. The uncertainly in the measurement of the displacement is . a.u. measured using Matlab Data Cursor tool The free space velocity and the numerical velocity of propagation are the same when = 1 (the time step is set equal to the magic time step). S Rectangular pulse = 0.95 S Fig. 2. Position of the rectangular pulse at fixed time intervals for 0.95. = 168 em_swe_03.m S = Nt Figure 2 shows the propagation of the pulse when 0.95 < 1. There is S = a ringing effect at the leading and trailing edges of the pulse. The numerical velocity is . The ringing represents time retarded propagation of the sparsely, high frequency spectral components of the discontinuities at the steps of the rectangular pulse. Rectangular pulse = 0.70 S Fig. 3. Position of the rectangular at fixed time intervals for 0.70. = 230 em_swe_03.m S = Nt If you carefully compare figures 1 and 3, there is a weak superluminal response () that propagates just ahead of the leading edge of the rectangular pulse. After the time interval of 8.00 a.u. the leading dt =
Rectangular pulse S = 1 magic time-step Fig. 1. Position of the rectangular pulse at fixed time intervals for S = 1 (magic time-step). Nt = 160 em_swe_03.m Figure 1 shows the position of the pulse after fixed time intervals when the time step is set equal to the magic time step where the Courant number is S = 1. The shape and spatial width of the pulse are completely preserved as it propagates along the X axis. The free space velocity is a.u. The numerical velocity vn can be measured from the displacement of the pulse dx from t = 0 a.u. and at the end of the execution time dt = 8.00 a.u. The uncertainly in the measurement of the displacement is . a.u. measured using Matlab Data Cursor tool The free space velocity and the numerical velocity of propagation are the same when S = 1 (the time step is set equal to the magic time step). Rectangular pulse S = 0.95 Fig. 2. Position of the rectangular pulse at fixed time intervals for S = 0.95. Nt = 168 em_swe_03.m Figure 2 shows the propagation of the pulse when S = 0.95 < 1. There is a ringing effect at the leading and trailing edges of the pulse. The numerical velocity is . The ringing represents time retarded propagation of the sparsely, high frequency spectral components of the discontinuities at the steps of the rectangular pulse. Rectangular pulse S = 0.70 Fig. 3. Position of the rectangular at fixed time intervals for S = 0.70. Nt = 230 em_swe_03.m If you carefully compare figures 1 and 3, there is a weak superluminal response () that propagates just ahead of the leading edge of the rectangular pulse. After the time interval of dt = 8.00 a.u. the leading
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