Adds comparison of filter for different order parameter

docs/make.jl

@@ -9,7 +9,7 @@ using
 ##### Generate examples
 #####
 
-@show bib_filepath = joinpath(@__DIR__, "src/references.bib")
+bib_filepath = joinpath(@__DIR__, "src/references.bib")
 bib = CitationBibliography(bib_filepath)
 
 const EXAMPLES_DIR = joinpath(@__DIR__, "..", "examples")

docs/src/timestepping.md

@@ -16,37 +16,46 @@ For example, `"RK4"` for the Runge-Kutta 4th-order time stepper.
 Most of the time steppers also come with their `Filtered` equivalents: [`FilteredForwardEulerTimeStepper`](@ref), [`FilteredAB3TimeStepper`](@ref), [`FilteredRK4TimeStepper`](@ref), [`FilteredLSRK54TimeStepper`](@ref), and [`FilteredETDRK4TimeStepper`](@ref).
 
 The filtered time steppers apply a high-wavenumber filter to the solution at the end of each step.
-The motivation behind filtering is to remove enstrophy accumulating at high wavenumbers and creating 
-noise at grid-scale level.
+The motivation behind filtering is to preclude enstrophy from accumulating at high wavenumbers and
+creating noise at grid-scale level.
 
-The high-wavenumber filter used in the filtered timesteppers is:
+The high-wavenumber filter used in the filtered time steppers is:
 
 ```math
 \mathrm{filter}(\boldsymbol{k}) = 
-     \begin{cases}
-       1 & \quad |\boldsymbol{k}| ≤ k_{\textrm{cutoff}} \, ,\\ 
-       \exp{ \left [- \alpha (|\boldsymbol{k}| - k_{\textrm{cutoff}})^p \right]} & \quad |\boldsymbol{k}| > k_{\textrm{cutoff}} \, .
-     \end{cases}
+  \begin{cases}
+    1 & \quad |\boldsymbol{k}| ≤ k_{\textrm{cutoff}} \, ,\\
+    \exp{ \left [- \alpha (|\boldsymbol{k}| - k_{\textrm{cutoff}})^p \right]} & \quad |\boldsymbol{k}| > k_{\textrm{cutoff}} \, .
+  \end{cases}
 ```
 
-For fluid equations with quadratic non-linearities it makes sense to choose a cutoff wavenumber
-at 2/3 of the highest wavenumber resolved in our domain, ``k_{\textrm{cutoff}} = \tfrac{2}{3} k_{\textrm{max}}`` (see discussion in [Aliasing section](@ref aliasing)).
+For fluid equations with quadratic non-linearities it makes sense to choose a cut-off wavenumber
+at 2/3 of the highest wavenumber that is resolved in our domain,
+``k_{\textrm{cutoff}} = \tfrac{2}{3} k_{\textrm{max}}`` (see discussion in [Aliasing section](@ref aliasing)).
 
-Given the order ``p``, we calculate the coefficient ``\alpha`` so that the the filter value
-that corresponds to the highest allowed wavenumber in our domain is a small value, ``\delta``,
-taken to be close to machine precision.
-
-That is:
+Given the order ``p``, we choose coefficient ``\alpha`` so that the filter value that corresponds
+to the highest allowed wavenumber in our domain is a small number ``\delta``, usually taken to be
+close to machine precision. That is:
 
 ```math
 \alpha = \frac{- \log\delta}{(k_{\textrm{max}} - k_{\textrm{cutoff}})^p} \ .
 ```
 
-The above filter originates from the book by [Canuto-etal-1987](@cite). In geophysical turbulence
-applications it was used by [LaCasce-1996](@cite) and later by [Arbic-Flierl-2004](@cite).
+The above-mentioned filter form originates from the book by [Canuto-etal-1987](@cite).
+In geophysical turbulence applications it was used by [LaCasce-1996](@cite) and later
+by [Arbic-Flierl-2004](@cite).
+
+!!! warning "Not too steep, not too shallow"
+    Care should be taken if one decides to fiddle with the filter parameters. Changing
+    the order ``p`` affects how steeply the filter falls off. Lower order values imply
+    that the filter might fall off too quickly and may lead to Gibbs artifacts; higher
+    order value implies that the filter might fall off too slow and won't suffice to
+    remove enstrophy accumulating at the grid scale.
 
-Using the default parameters provided by the filtered time steppers (see
-[`FourierFlows.makefilter`](@ref)), the filter has the following form:
+The filter using the default parameters provided by the filtered time steppers (see
+[`FourierFlows.makefilter`](@ref)) is depicted below. The same plot also compares how
+the filter changes when we vary the order parameter ``p`` while keeping everything
+else the same.
 
 ```@setup 1
 using CairoMakie
@@ -55,16 +64,21 @@ set_theme!(Theme(linewidth = 3, fontsize = 20))
 ```
 
 ```@example 1
-using FourierFlows, CairoMakie
-
-K = 0:0.01:1 # non-dimensional wavenumber k * dx / π
+using CairoMakie
+using FourierFlows: makefilter
 
-filter = FourierFlows.makefilter(K)
+K = 0:0.001:1 # non-dimensional wavenumber k * dx / π
 
 fig = Figure()
-ax = Axis(fig[1, 1], xlabel = "k dx / π", ylabel = "filter", aspect=2.5, xticks=0:0.2:1)
+ax = Axis(fig[1, 1], xlabel = "|𝐤| dx / π", ylabel = "filter", aspect=2.5, xticks=0:0.2:1)
+
+vlines!(ax, 2/3, color = (:gray, 0.4), linewidth = 6, label = "cutoff wavenumber |𝐤| dx / π = 2/3 (default)")
+
+lines!(ax, K, makefilter(K), linewidth = 4, label = "order 4 (default)")
+lines!(ax, K, makefilter(K, order = 1), linestyle = :dash, label = "order 1")
+lines!(ax, K, makefilter(K, order = 10), linestyle = :dot, label = "order 10")
 
-lines!(ax, K, filter)
+axislegend(position = :lb)
 
 current_figure() # hide
 ```

src/domains.jl

@@ -485,7 +485,7 @@ one-dimensional grid, the non-dimensional wavenumber `K` is
 K = k * dx / π
 ```
 
-and thus it takes values `K` ``∈ [-1, 1]``.
+and thus take values in ``[-1, 1]``.
 
 For `K ≤ innerK` the filter is inactive, i.e., equal to 1. For `K > innerK`,
 the filter decays exponentially to remove high-wavenumber content from 
@@ -495,8 +495,9 @@ the solution, i.e.,
 filter(K) = exp(- decay * (K - innerK)^order)
 ```
 
-For a given `order` and , the `decay` rate is determined so that the filter value at the
-outer wavenumber `outerK` is `tol`, where `tol` is a small number, close to machine precision.
+For a given `order`, the `decay` rate is determined so that the filter value at the
+outer wavenumber `outerK` is `tol`, where `tol` is a small number, close to machine
+precision.
 
 ```julia
 decay = - log(tol) / (outerK - innerK)^order