diff --git a/previews/PR387/barotropic/index.html b/previews/PR387/barotropic/index.html index 02df96940..77882368a 100644 --- a/previews/PR387/barotropic/index.html +++ b/previews/PR387/barotropic/index.html @@ -9,4 +9,4 @@ w_{i+1} &= u_{i-1} + 2\Delta tF(v_i) \\ u_i &= v_i + \frac{\nu\alpha}{2}(w_{i+1} - 2v_i + u_{i-1}) \\ v_{i+1} &= w_{i+1} - \frac{\nu(1-\alpha)}{2}(w_{i+1} - 2v_i + u_{i-1}) -\end{aligned}\]
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.
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This document was generated with Documenter.jl version 0.27.25 on Monday 16 October 2023. Using Julia version 1.8.5.