Other Damping Stencils
The a = 0.3 n 7-point damping stencil with damping function shown in Figure 7.2 is primarily designed for problems with discontinuities or shocks. Because of the steep gradients involved, strong damping over a wide band of wave numbers is necessary. But even in problems for which the solutions are shock and discontinuity free, short spurious waves are often generated. Wall boundaries, mesh size change interfaces and regions with steep gradients are potential sources of short waves. For this reason, it is a good practice to add a small amount of background artificial selective damping to a computation. Also because of the need to use backward difference stencils at the boundary region of a computation domain, the numerical solution is subjected to weak numerical instability. These instabilities usually manifest themselves in the form of grid-to-grid oscillations with very gradual increase in amplitude. This type of weak instability can easily be suppressed by the addition of artificial selective damping.
For general background damping for which only short waves need to be removed, the following a = 0.2 n 7-point damping stencil is recommended. The stencil coefficients are as follows:
7-Point Damping Stencil (a = 0.2n)
d0 = 0.2873928425 d4 = d_! = -0.2261469518 d2 = d_2 = 0.1063035788 d3 = d_3 = -0.0238530482
The damping function of this damping stencil is shown in Figure 7.4.
When very high resolution is required, a 15-point stencil DRP scheme may be used. The following is a set of damping coefficients for such a stencil.
15-Point Damping Stencil
d0 = 0.2042241813072920 d1 = d_1 = -0.1799016298200503 d2 = d_2 = 0.1224349282118140 d3 = d_3 = -6.3456279827554890E – 02 d4 = d_4 = 2.4341225689340974E – 02 d5 = d_5 = -6.5519987489327603E – 03 d6 = d_6 = 1.1117554451990776E – 03 d7 = d_7 = -9.0091603462069583E – 05
The damping curve is shown in Figure 7.4.