tutorial.jl

### A Pluto.jl notebook ###
# v0.19.29

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 3c82911b-5edc-4148-a0b4-ddff71e1d6d7
begin
	using Plots
	using LaTeXStrings
	using PlutoUI
	using Random
	TableOfContents()
end

# ╔═╡ 9ea6a24c-6616-11ee-1314-859b895875ec
md"## Scientific Computation with Julia (Fluids 9808)"

# ╔═╡ 542449e5-e8d8-4ab5-8f72-b1b89ebaa0be
md"## Part 0: Pluto Notebooks

Pluto is similar to Jupyter notebooks, but with a few nice features:
1. **Reproducible** — can share code with others, knowing it'll run the same, with the same package versions, etc. 

2. **Reactive** — you're able to save a bunch of time when prototyping + the notebook always showcases the current state of the program, no need to worry about running the cells out of order. encourages you to write _safe_ code. 

and a bunch of nice touches!
* Table of Contents
* Live Docs
* LaTeX Rendering
* Hiding code cells 
* Sharing flexibility

... [output above the cells](https://github.com/fonsp/Pluto.jl/issues/205)"

# ╔═╡ 96d612c0-0b33-412e-ad82-d2cf4535ec64
md"## Part 1: A brief intro to Julia"

# ╔═╡ c76681c7-efe4-465f-8f1e-2928d41819e7
md"### Vectors & Matrices"

# ╔═╡ f2e5e54c-6f6a-4a33-b429-01a3110287b6
begin
	a = [1 2 3]
	b = [4 5 6]
end

# ╔═╡ 9ff5cd65-71cf-45f5-8097-e5db66f2245d
md"Concatenation is super easy!"

# ╔═╡ 2fc9579e-ea67-4226-94a5-b633ed5e51ed
A = [a; b]

# ╔═╡ 893161b0-9a52-4f3f-a024-feb62d90011d
md"Julia is 1-indexed: (yay?)"

# ╔═╡ 4243aedb-4325-49b1-af8d-676dd2cdeaa1
A[1,2]

# ╔═╡ 96d6da16-d95d-4a92-b175-9c163aacfec3
md"Note that most of the function names/calls are the same as python/matlab:"

# ╔═╡ 9f3c60fb-6fe4-4d6c-acf6-650bad315da4
zeros(8,8)

# ╔═╡ 908d2130-a3ee-49fb-a1fa-88c8755725d2
md"We often want to create vectors for scientific computations:"

# ╔═╡ 260d1fb1-f320-4974-a69f-3229e4ffaf6c
collect(range(0,10,1000))

# ╔═╡ ffe83899-ab25-484c-a5e9-b082d00afa45
typeof(collect(0:0.1:1))

# ╔═╡ c0dd1e0a-9660-4c49-bada-b9b02c9828c0


# ╔═╡ 38323b77-fa3f-46af-9f03-1ce022ace4c6
md"### Control Flow, Loops & Functions"

# ╔═╡ 9bbd6d27-1570-45c3-9bea-414f61e99aa9
function fibonacci(n::Int)
    if n <= 0
        return []
    elseif n == 1
        return [0]
    elseif n == 2
        return [0, 1]
    else
    	fibs = [0, 1]
    	for i in 3:n
            push!(fibs, fibs[end] + fibs[end-1])
        end
        return fibs
    end
end

# ╔═╡ 8b78aea8-2b1c-4bcf-9e0a-7d4ecbc1fcee
fibonacci(10)

# ╔═╡ 48efd8bb-256d-4610-a4b0-277a5a2d21bb


# ╔═╡ 7159ea65-e068-47d4-99fe-cfed686cab07
md"### Plotting 
(& some super-cool Pluto interactivity)"

# ╔═╡ e8774533-a28a-457f-9fd0-b971320e4b93
md"The plotting package in Julia has a bit of a secret... it's really ~_all_ of the plotting packages in one. 

''[_Don't commit to one graphics package. Use the same code and access the strengths of all backends._](https://docs.juliaplots.org/latest/backends/)'' "

# ╔═╡ cb80f9cf-5f93-4896-84b5-6a285cb3484b
md"Sidenote: [types in Julia](https://en.wikibooks.org/wiki/Introducing_Julia/print#/media/File:Julia-number-type-hierarchy.svg) are actually quite easy to understand :)"

# ╔═╡ e0225f73-f324-4624-8bfd-78f9fdcd3b75
typeof(π)

# ╔═╡ cfc63ec8-d658-4eac-ab46-440fd7f3a65b
gr()

# ╔═╡ 7ea15a1b-6304-4b32-9244-ed2823e4f414
scatter(randn(10))

# ╔═╡ 620bbf63-4589-4d67-a03a-2edff1655fbf
@bind γ PlutoUI.Slider(0.0:0.01:1.0, default=0.15)

# ╔═╡ 9a4d1050-7c7e-4c78-9001-c1a19d5751a5
begin
	z = collect(range(0,2*π,100))
	s = sin.(z)
	sNoise = s .+ γ.*randn(length(s))
end

# ╔═╡ 1ae6f251-f179-4c67-9752-54a0065491b5
md"γ = $(γ)"

# ╔═╡ fb13f5fc-31d0-459a-9635-c807f0d4d621
begin
	plot(z, s, lw = 3, label = "sin(x)", legend=:topright)
	scatter!(z, sNoise, markersize = 2, label = "with noise")
	xlabel!("z")
	ylims!(-1.5,1.5)
end

# ╔═╡ 70ab0515-6376-4ef4-ac62-245f503ff6f4


# ╔═╡ 6d4cc44e-6654-47ff-81ef-ce6a6912b9f9
md"#### Subplots

Subplots are really just vectors of plot series."

# ╔═╡ 576d3c4c-6bb6-4106-8be3-fed27ee8cd9a
begin
	w = collect(range(0,10,1000))
	
	y1 = @. exp(-0.1w) * cos(4w)
	y2 = @. exp(-0.3w) * cos(4w)
	y3 = @. exp(-0.5w) * cos(4w)
end

# ╔═╡ c0197395-5778-4579-912e-7c418ff53f87
begin
	# Method 1
	# plot(w, [y1 y2 y3], layout=(3, 1), legend=false)
	# xlabel!("x")

	# Method 2
	y1plot = plot(w, y1, label = L"$\alpha = -0.1$")
	y2plot = plot(w, y2, label = L"$\alpha = -0.3$")
	y3plot = plot(w, y3, label = L"$\alpha = -0.5$")

	plot(y1plot, y2plot, y3plot, layout=(3, 1))
	xlabel!("x");
end

# ╔═╡ 8b69f2a7-7ecc-48ba-be1e-5c1e54dc9d44
md"[This is really just scratching the surface of the Plots package!](https://docs.juliaplots.org/latest/)"

# ╔═╡ 28fe8c33-fd06-4784-bee5-80be8ff75c8c
md"## Part 2: Solving the advection-diffusion eq"

# ╔═╡ e04fc39f-0505-4262-bdc4-614022a7343f
md"The advection-diffusion equation is

$$\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + D \frac{\partial^2 T}{\partial t^2} = 0, \ \ \text{for} \ \  0 \leq x \leq L$$

where $u$ is the advective velocity and $D$ is the diffusivity constant."

# ╔═╡ 67f20b40-dde8-4c06-843d-844bd3762f22
begin
	u = 1.0
	D = 0.01
	
	Nx = 100
	L = 1
		
	x = collect(range(0,L, Nx)) 
	Δx = x[2] - x[1]

	if D == 0
		Δt = Δx / u
	else
		Δt = 0.5 * (Δx^2)/D # u = 0 condition
	end
	
	t = collect(0:Δt:10)
	Nt = length(t)
end

# ╔═╡ 18f751d1-80bf-429d-aefc-bc25cd276fd6
md"Define initial condition:"

# ╔═╡ b436c9d6-4107-4eef-8b39-faadfd23ba02
begin
	# initialize solution array 
	T = zeros(Nt,Nx)

	# create initial condition and plot 
	# T[1,:] = sin.(2*π*x) .+ 0.5  
	T[1,:] = sin.(4*π*x) .+ 0.5 .+ sin.(2*π*x) .+ 0.5  

	line = plot(x, T[1,:], label = L"$T_o (x)$", linecolor = :orange, lw = 2)
	xlabel!("x"); ylabel!("T(x)");

	heat = heatmap(x, [0.], collect(T[1,:]'), 
			   clims=(-1., 1.), cbar=false, xticks=nothing, yticks=nothing)

	plot(line, heat, layout = grid(2, 1, heights=[0.9, 0.1]), clim=(-1, 1))
end

# ╔═╡ 42a9b881-ad72-4bcb-bf66-ebdab9449990
md"Now let's solve the PDE for this initial condition. The finite difference scheme is given by: 

$$T_i^{n+1} = T_i^n - \alpha(T_{i+1}^n - T_{i-1}^n) + \beta (T_{i+1}^n - 2T_{i}^n + T_{i-1}^n)$$ where

$$\alpha = u \frac{\Delta t}{2 \Delta x} \ \ \text{and} \ \ \beta = D\frac{\Delta t}{\Delta x^2}$$"

# ╔═╡ 9f88b6cd-c3d0-4a73-82ac-aed2197a6112
begin
	α = u * (Δt / (2 * Δx))
	β = D * (Δt / (Δx^2))

	if D == 0 # backward difference
		for n = 1:(Nt-1)
				
			for i = 2:(Nx-1)
					
					T[n+1,i] = T[n,i] - 2*α*(T[n,i] - T[n,i-1]) + β*(T[n,i+1] - 2*T[n,i] + T[n,i-1])
					
			end
		
			T[n+1, 1] = T[n,1] - 2*α*(T[n,1] - T[n,end]) + β*(T[n,2] - 2*T[n,1] + T[n,end])
				
			T[n+1,end] = T[n,end] - 2*α*(T[n,end] - T[n,end-1]) + β*(T[n,1] - 2*T[n,end] + T[n,end-1])
				
		end
	else # centered difference
		for n = 1:(Nt-1)
			
			for i = 2:(Nx-1)
				
				T[n+1,i] = T[n,i] - α*(T[n,i+1] - T[n,i-1]) + β*(T[n,i+1] - 2*T[n,i] + T[n,i-1])
				
			end
	
			T[n+1, 1] = T[n,1] - α*(T[n,2] - T[n,end]) + β*(T[n,2] - 2*T[n,1] + T[n,end])
			
			T[n+1,end] = T[n,end] - α*(T[n,1] - T[n,end-1]) + β*(T[n,1] - 2*T[n,end] + T[n,end-1])
			
		end
	end
	
end

# ╔═╡ ecbb5776-fbcc-4a03-a08f-502159c5c516
md"(timestep) n = $(@bind timepoint PlutoUI.Slider(1:Nt, show_value = true, default=1))"

# ╔═╡ f8854ea8-acd5-4343-b208-e81e1826e8e6
begin
	p1 = plot(x,T[timepoint,:], legend = false)
	scatter!(x,T[timepoint,:])
	# plot!(x,T[1,:], label = L"T_o (x)")
	ylabel!("T"); xlabel!("x")
	ylims!(-0.8,2.8)

	p2 = heatmap(x, [0.], collect(T[timepoint,:]'), 
			   clims=(-1., 1.), cbar=false, xticks=nothing, yticks=nothing)

	plot(p1, p2, layout = grid(2, 1, heights=[0.9, 0.1]), clim=(-1, 1))
end

# ╔═╡ 88a4ee12-5f57-48fd-a544-274ca53789f4
md"## Other Resources:

* MIT Course: [Intro to Computational Thinking](https://computationalthinking.mit.edu/Fall23/)
* [Whirlwind intro to Julia](https://www.youtube.com/watch?v=8h8rQyEpiZA&t=7898s)
* [Julia Documentation](https://docs.julialang.org/en/v1/) 


* [Package manager overview](https://www.youtube.com/watch?v=76KL8aSz0Sg) + [Docs](https://pkgdocs.julialang.org/v1/)
* [Pluto Notebooks FAQ](https://github.com/fonsp/Pluto.jl/wiki)

* [Pluto notebooks](https://www.youtube.com/watch?v=IAF8DjrQSSk) (talk)
* [The Two Cultures of Programming](https://www.youtube.com/watch?v=C3iR_PknlFc) (talk)
* [Multiple dispatch](https://www.youtube.com/watch?v=kc9HwsxE1OY&t=1544s) (talk)
* [What Julia Offers Academic Researchers](https://www.youtube.com/watch?v=C4dMYHzW-SY&t=443s) (talk)
* [Wrote a textbook using Julia](https://www.youtube.com/watch?v=ofWy5kaZU3g) (talk)
* [Julia for Biologists](https://www.youtube.com/watch?v=gRj7E5kYG1I&t=1937s) (talk)"

# ╔═╡ 00000000-0000-0000-0000-000000000001
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Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[compat]
LaTeXStrings = "~1.3.0"
Plots = "~1.39.0"
PlutoUI = "~0.7.52"
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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