### A Pluto.jl notebook ###
# v0.19.27

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

# ╔═╡ c223c2ae-4ddc-4a5c-b66b-561724c6c01d
using LinearAlgebra, Plots, LaTeXStrings

# ╔═╡ 34f0cbe9-00b3-4e50-a097-054770a27eff
using SparseArrays

# ╔═╡ 9b552053-cd6e-463c-946b-da4c7a668a5a
using Printf

# ╔═╡ de2b4bee-ca03-47e6-9b5c-be3bf7f87bb4
using PlutoUI

# ╔═╡ bcec0bba-2f5d-49d7-a0d3-e4ba50afabc0
using OrdinaryDiffEq

# ╔═╡ 176d6bba-933b-4904-84be-92aef76250ac
using SummationByPartsOperators

# ╔═╡ 5b4bf62a-ece2-4f2e-b506-be744deedb5f
md"""
# A brief tour of Julia
"""

# ╔═╡ ede66c45-4b31-4a62-a312-e635db7646b1
md"""
## Unicode and LaTeX
"""

# ╔═╡ 69cea251-ee9d-44dd-9714-f7a0e60601f1
β = π / 4 # type `\beta` + TAB

# ╔═╡ a4090148-0ebc-48fc-b98a-8888570c2da2
tan(β) ≈ sin(β) / cos(β)

# ╔═╡ 3ffd2a9f-eb6a-4ae3-9399-524a87617a24
begin
	λ = eigvals(randn(1_000, 1_000))
	scatter(real(λ), imag(λ), aspect_ratio = :equal, legend = nothing,
        xguide = L"\operatorname{Re} \lambda", 
		yguide = L"\operatorname{Im} \lambda")
end

# ╔═╡ 97dee24c-73e4-4186-a1a4-ea4e4c8a6104
md"""
## Linear algebra (sparse and dense)
"""

# ╔═╡ 15ce050a-02d0-4ade-be57-84c6568ed7e3
begin
	n = 30
	A = spdiagm(0 => 2 * ones(n), 1 => -ones(n-1), -1 => -ones(n-1))
end

# ╔═╡ d44daaa1-701c-4a4d-930d-76f9fcff8e6d
begin
	f = rand(n)
	u = A \ f
	norm(A * u - f)
end

# ╔═╡ 49013e3e-4a23-11ed-3281-85e869263467
md"""
## Wave equation

Consider the wave equation

$$\begin{equation*}
  \begin{aligned}
    \partial_t^2 u(t, x) &= \partial_x^2 u(t, x) && \text{in } (0,T) \times (x_\mathrm{min}, x_\mathrm{max}), \\
		\partial_x u(t, x) &= 0 && \text{on } (0, T) \times \{x_\mathrm{min}, x_\mathrm{max}\}, \\
    u(t, x) &= u^0(x) &&\text{on } \{0\} \times [x_\mathrm{min}, x_\mathrm{max}], \\
    \partial_t u(t, x) &= v^0(x) &&\text{on } \{0\} \times [x_\mathrm{min}, x_\mathrm{max}].
  \end{aligned}
\end{equation*}$$

Here, we choose the initial conditions

$$\begin{equation*}
\begin{aligned}
  u^0(x) &= \exp(-20 x^2), \\
  v^0(x) &= 0, \\
\end{aligned}
\end{equation*}$$

with parameters $x_\mathrm{min} = -1$, $x_\mathrm{max} = 1$.

We use the classical second-derivative SBP operator
(central differences in the interior, one-sided differences at the boundaries)
in space and a third-order accurate Runge-Kutta method in time.
The parameters are
- ``N``: number of grid points distributed uniformly in the domain
- ``\Delta x``: grid spacing
- ``\Delta t``: time step size

"""

# ╔═╡ 27b7b87b-5d76-4893-9d32-9b1fd7d200b1
md"""
You should play around with the parameter settings above. In particular, you
should have a look at the following questions:

- How does the solution evolve in time? You can choose the final time ``T``
  to investigate this behavior.
- How does the error behave under grid refinement, i.e., increasing ``N``?
- How do you need to choose the time step size ``\Delta t`` depending on the
  spatial resolution ``N \propto 1 / \Delta x``?
"""

# ╔═╡ 823036b2-f41d-4642-995f-4e96cc6f9f15
md"""
# Appendix

You can find code and utility material in this appendix.
"""

# ╔═╡ e3c5952c-2125-416e-a718-ccf5db02933a
space = html"<br><br><br>";

# ╔═╡ 2b28849d-c3d4-4f69-80c7-60ce5cb6f487
space

# ╔═╡ 417e5583-7d99-4eac-953f-b84cce3a72b0
space

# ╔═╡ 668f1ccd-7e7a-4d6e-b3f5-23a8389244e2
space

# ╔═╡ 7a3580a3-3cb2-4282-a9f3-b32702d8c8f7
# ╠═╡ show_logs = false
# ╠═╡ disabled = true
#=╠═╡
using LaTeXStrings
  ╠═╡ =#

# ╔═╡ 550930ed-7815-46a5-b228-7c1eba2aa614
# ╠═╡ disabled = true
#=╠═╡
using Plots: Plots, plot, plot!, scatter, scatter!
  ╠═╡ =#

# ╔═╡ de68c071-b710-498b-abb4-2b729250e09a
function rhs!(ddu, du, u, parameters, t)
	D2, = parameters
	mul!(ddu, D2, u)
	ddu[begin] += derivative_left(D2, u, Val{1}()) / left_boundary_weight(D2)
	ddu[end] -= derivative_right(D2, u, Val{1}()) / right_boundary_weight(D2)
	return nothing
end

# ╔═╡ 982e9af9-ba93-4855-bb0a-a3bd4249ab2c
# We can interpret the solution of the problem with homogeneous Neumann BCs
# as the superposition of multiple wave packets on the whole real line.
# We use that the initial condition is essentially zero at the boundaries
# (in 64 bit floating point arithmetic).
function usol(t, x)
	u = zero(t + x)
	# This is only accurate up to time 10
	for i in 0:5
		left  = 0.5 * exp(-20 * (x - t + 2 * i)^2)
		right = 0.5 * exp(-20 * (x + t - 2 * i)^2)
		u = u + left + right
	end
	return u
end

# ╔═╡ dd98c2bd-b0b9-4c73-8ff7-e66d31c05204
const Tmax = 6.0

# ╔═╡ 415276e3-6bf4-4337-986d-820ec282ea0b
md"""
``N`` = $(@bind N Slider([20, 40, 80, 160], default=40, show_value=true))

(``\Delta x`` = $(1 / N))

``\Delta t`` = $(@bind Δt Slider([1.0e-3, 5.0e-3, 1.0e-2, 2.0e-2, 4.0e-2, 8.0e-2, 0.16], default=1.0e-2, show_value=true))

``T`` = $(@bind T Slider(0.0:0.02:Tmax, default=0.2, show_value=true))
"""

# ╔═╡ 80620afb-7a5f-4511-9afd-d80c9d045885
begin
	D2 = derivative_operator(MattssonNordström2004(),
						     derivative_order = 2, accuracy_order = 2,
						     xmin = -1.0, xmax = 1.0, N = N)
	x = SummationByPartsOperators.grid(D2)
	u0 = @. exp(-20 * x^2)
	v0 = zero(u0)
	nothing
end

# ╔═╡ 13e33aa8-f7b4-4a28-b6ff-9c4351af361c
begin
	ode = SecondOrderODEProblem(rhs!, v0, u0, (0.0, Tmax), (D2,))
	sol = solve(ode, SSPRK33(); adaptive = false, dt = Δt)
	nothing
end

# ╔═╡ 46f8b441-af8f-40e3-9856-acfd69d70c20
let
	x = SummationByPartsOperators.grid(D2)
	err = integrate(abs2, usol.(T, x) - sol(T).x[2], D2) |> sqrt
	md"""
	Error at the time ``T``: $(@sprintf("%.2e", err))
	"""
end

# ╔═╡ 0a071947-9e23-4218-9436-915f08e1a83c
begin
	fig = plot(xguide = L"x", yguide = L"u", thickness_scaling = 1.5,
			   legend = :outertopright, size = (800, 400))
	x_plot = range(-1.0, 1.0, length = 200)
	plot!(x_plot, usol.(first(ode.tspan), x_plot), label = L"u^0")
	scatter!(fig, x, sol(T).x[2], label = L"u^\mathrm{num}")
	plot!(fig, x_plot, usol.(T, x_plot), label = L"u^\mathrm{ana}")
	min, max = extrema(sol.u[end].x[2])
	if min > -0.1 && max < 1.1
		plot!(fig, yrange = (-0.1, 1.1))
	end
	fig
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
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[compat]
LaTeXStrings = "~1.3.0"
OrdinaryDiffEq = "~6.51.1"
Plots = "~1.38.8"
PlutoUI = "~0.7.50"
SummationByPartsOperators = "~0.5.30"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.3"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
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 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

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