with the Williams filter parameter $\alpha \in [0.5,1]$. For $\alpha=1$ we're back with the Robert-Asselin filter (the first two lines).
The Laplacian in the parentheses is often called a displacement, meaning that the filtered value is displaced (or corrected) in the direction of the two surrounding time steps. The Williams filter now also applies the same displacement, but in the opposite direction to the next time step $i+1$ as a correction step (line 3 above) for a once-filtered value $v_{i+1}$ which will then be twice-filtered by the Robert-Asselin filter on the next iteration. For more details see the referenced publications.
The initial Euler step (see Time integration, Table) is not filtered. Both the the Robert-Asselin and Williams filter are then switched on for all following leapfrog time steps.
Robert66Robert, André. “The Integration of a Low Order Spectral Form of the Primitive Meteorological Equations.” Journal of the Meteorological Society of Japan 44 (1966): 237-245.
Williams2009Williams, P. D., 2009: A Proposed Modification to the Robert–Asselin Time Filter. Mon. Wea. Rev., 137, 2538–2546, 10.1175/2009MWR2724.1.
Amezcua2011Amezcua, J., E. Kalnay, and P. D. Williams, 2011: The Effects of the RAW Filter on the Climatology and Forecast Skill of the SPEEDY Model. Mon. Wea. Rev., 139, 608–619, doi:10.1175/2010MWR3530.1.
Williams2011Williams, P. D., 2011: The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations. Mon. Wea. Rev., 139, 1996–2007, doi:10.1175/2010MWR3601.1.
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