Central Difference Scheme with Spatial Smoothing 



The central difference scheme contains less numerical diffusion than the 

 upwind scheme but often produces numerical noise in the form of grid-to-grid 

 oscillation, particularly when the grid points are too sparse to adequately 

 resolve a given change in a variable. In the absence of strong physical 

 damping, growth of these short-wave oscillations seriously deteriorates the 

 numerical solution and may even lead to instability. We have designed a 

 spatial smoothing scheme (Sheng, Segur, and Lewellen, 1978) to damp these 

 short-wave oscillations out of the numerical solution while maintaining 

 reasonable accuracy in the background solution. 



Given a solution profile at an instant of time, the smoothing scheme 

 first chests for the slopes and curvatures at each point. For example, a 

 typical profile of a one-dimensional variable V containing short-wave 

 oscillations is shown in Figure E.2. The profile at a typical point C is 

 considered to contain a peak (no smoothing applied otherwise) if: 



^ + A,_ > p/^ (E-5) 



where 



^ = |Vj.l - Vj|/Ax 



and u is a constant not smaller than 2. If Eq. (E-5) is satisfied, the 

 curvature at C is then compared with those at two neighboring points to see if 

 the peak is associated with short-wave or long-wave oscillations. It is 

 considered to be a short-wave oscillation if: 



A^ X A^ < (E-6) 



R 



or 



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