From bac4de980a5265322713a4fe3b27f0a8c86b9750 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andr=C3=A9s=20Rueda-Ram=C3=ADrez?= <aruedara@uni-koeln.de>
Date: Thu, 16 May 2024 17:41:23 +0200
Subject: [PATCH] Added notebook to explain the FD methods for the
 advection-diffusion equation

---
 ...e_difference_method_advection_diffusion.jl | 1317 +++++++++++++++++
 1 file changed, 1317 insertions(+)
 create mode 100644 notebooks/finite_difference_method_advection_diffusion.jl

diff --git a/notebooks/finite_difference_method_advection_diffusion.jl b/notebooks/finite_difference_method_advection_diffusion.jl
new file mode 100644
index 0000000..750e926
--- /dev/null
+++ b/notebooks/finite_difference_method_advection_diffusion.jl
@@ -0,0 +1,1317 @@
+### A Pluto.jl notebook ###
+# v0.19.40
+
+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
+
+# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276
+using Plots, PlutoUI, LaTeXStrings, ShortCodes, LinearAlgebra
+
+# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e
+md"""
+# Finite differences for advection-diffusion
+
+In this notebook, we will approximate the solution to the steady state advection-diffusion equation in one dimension,
+```math
+\begin{equation}
+- \kappa \frac{\partial^2 u}{\partial x^2} + b \frac{\partial u}{\partial x} = f,
+\end{equation}
+```
+using a **finite difference method**. Here, ``u(x)`` is the temperature, ``x`` is the spatial dimension, ``\kappa`` is the thermal conductivity, ``b`` is the advection velocity, and ``f(x)`` is a known heat source. For simplicity, we will assume $b = 1$ in the following sections and drop it from the equations. 
+
+Our computational domain is $\Omega = [0, 1]$, and we assume $u$ to be zero on the boundaries, i.e., $u(0) = u(1) = 0$. We discretize our domain with $N + 1$ nodes. The nodes $\{ x_i \}$ with $i = 0, \ldots, N$ are equidistantly placed and their positions are computed as
+```math
+\begin{align}
+x_i &= ih,
+\end{align}
+```
+where the $h = 1/N$ is the distance between the nodes.
+
+To discretize the advection-diffusion equation, we use second-order central finite differences to approximate the second-order partial derivative,
+```math
+\frac{\partial^2 u(x)}{\partial x^2} \approx \frac{u(x-h) - 2 u(x) + u(x+h)}{h^2},
+```
+and three different discretization schemes for the first-order derivative:
+* Central difference:
+```math
+\frac{\partial u(x)}{\partial x} \approx \frac{u(x+h) - u(x-h)}{2 h},
+```
+
+* Upwind difference:
+```math
+\frac{\partial u(x)}{\partial x} \approx \frac{u(x) - u(x-h)}{h},
+```
+
+* Downwind difference:
+```math
+\frac{\partial u(x)}{\partial x} \approx \frac{u(x + h) - u(x)}{h},
+```
+
+Inserted in the advection-diffusion equation, we obtain
+* For the central difference:
+```math
+\begin{align}
+-\kappa \frac{u_{i-1} - 2 u_{i} + u_{i+1}}{h^2} + \frac{u_{i+1} - u_{i-1}}{2h} &= f_{i},\quad i = 1, \ldots, N-1,
+\end{align}
+```
+* For the upwind difference:
+```math
+\begin{align}
+-\kappa \frac{u_{i-1} - 2 u_{i} + u_{i+1}}{h^2} + \frac{u_{i} - u_{i-1}}{h} = f_{i},\quad i = 1, \ldots, N-1,
+\end{align}
+```
+* For the downwind difference:
+```math
+\begin{align}
+-\kappa \frac{u_{i-1} - 2 u_{i} + u_{i+1}}{h^2} + \frac{u_{i+1} - u_{i}}{h} = f_{i},\quad i = 1, \ldots, N-1,
+\end{align}
+```
+where $u_{i} = u(x_i)$ is the value of the temperature at node ``i``, and $f_{i} = f(x_i)$. 
+"""
+
+# ╔═╡ f87f000f-e0c9-46eb-a242-89a4290bd84f
+md"""
+### Numerical solution
+In the discretized equation $(*)$ above, each temperature value $u_{i}$ depends on its neighboring values. Thus, we end up with a linear system,
+```math
+A u = b
+```
+which we need to solve to obtain the temperature values at each inner node. Here, $A$ is the coefficient matrix, $u$ contains our unknown temperature values $u_{i}$ that we want to solve for, and $b$ holds the right-hand side of the system, i.e., $f_{i}$.
+
+We will solve the system using one of Julia's built-in linear solvers from the `LinearAlgebra` module. For simplicity, we will use $f(x) = 2$ everywhere:
+"""
+
+# ╔═╡ a7e57f8e-2032-419f-b957-12e90c3b9773
+md"""
+In the plot below, you can see a surface plot of the numerical solution, where the height indicates the value of the temperature at a given point. Increasing the number of intervals $N$, and thus decreasing the spacing between the nodes $h$, will lead to longer computing times but also a more accurate solution.
+"""
+
+# ╔═╡ 138b0481-7d42-4197-8ca2-b3fb073366a0
+md"""
+### Assembling the linear system
+From equation $(*)$ above, we can assemble our coefficient matrix $A$ that represents our linear system of equations (see class notes):
+"""
+
+# ╔═╡ 36f8657b-85db-4639-9941-6f7888a30b81
+"""
+fd_central(kappa, h, m)
+
+Constructs the system matrix for the advection-diffusion equation using a central second-order FD discretization of the second derivative and a central second-order FD for the first derivative.
+"""
+function fd_central(kappa, h, m)
+	# Banded matrix with full bands
+	A = diagm(    0  => fill(2 * kappa / h^2, m),
+	              1  => fill((1 - 2 * kappa / h) / (2 * h), m - 1),
+		         -1  => fill((-1 - 2 * kappa / h) / (2 * h), m - 1),
+	)
+	return A
+end
+
+# ╔═╡ 73c82ffa-a5e0-4040-8525-cc69a36a880d
+"""
+fd_upwind(kappa, h, m)
+
+Constructs the system matrix for the advection-diffusion equation using a central second-order FD discretization of the second derivative and an upwind first-order FD for the first derivative.
+"""
+function fd_upwind(kappa, h, m)
+	# Banded matrix with full bands
+	A = diagm(    0  => fill((2 * kappa / h + 1) / h, m),
+	              1  => fill(-kappa / h^2, m - 1),
+		         -1  => fill((-kappa / h - 1) / h, m - 1),
+	)
+	return A
+end
+
+# ╔═╡ 336dd44e-1791-4aca-9258-ed958f704892
+"""
+fd_downwind(kappa, h, m)
+
+Constructs the system matrix for the advection-diffusion equation using a central second-order FD discretization of the second derivative and a downwind first-order FD for the first derivative.
+"""
+function fd_downwind(kappa, h, m)
+	# Banded matrix with full bands
+	A = diagm(    0  => fill((2 * kappa / h - 1) / h, m),
+	              1  => fill((-kappa / h + 1) / h, m - 1),
+		         -1  => fill(-kappa / h^2, m - 1),
+	)
+	return A
+end
+
+# ╔═╡ 25a2120b-e9cd-48bc-901e-5611a2330f99
+md"""
+First-order Finite Difference: ``f = ``$(@bind selected_discretization Select([fd_central, fd_upwind, fd_downwind]))
+
+Number of intervals ``N``: $(@bind N Slider(4:1000, default=5, show_value=true))
+
+Heat conductivity ``\kappa``: $(@bind kappa Slider([1e-5, 1e-4, 1e-3, 1e-2, 1e-1, 1], default=1, show_value=true)) (logarithmic scale)
+"""
+
+# ╔═╡ f185aacd-0334-4217-8fa9-9fd0776c12c6
+begin
+	h = 1/N # Grid size
+	m = (N - 1) # System size
+
+	A = selected_discretization(kappa, h, m)
+	b = ones(m) # build right-hand side
+	u = A \ b # solve linear system
+	u = [0; u; 0] # Boundary conditions
+	x = 0:h:1
+	nothing # hide output
+end
+
+# ╔═╡ eb184f17-09c4-426e-80bb-f7d5867a01a4
+begin
+	#surface(reshape(u, (N - 1, N - 1)), camera=(angle, 45), zlims=(0.0, 0.15)) # Plot
+	x_fine = 0:0.001:1
+	if kappa > 0.01
+		exact_sol = x_fine .- (1 .- exp.(x_fine ./ kappa)) ./ (1 - exp(1 / kappa))
+	else
+		exact_sol = x_fine .- exp.((x_fine .- 1) ./ kappa)
+	end
+	plot(x_fine, exact_sol, label="Exact solution", linewidth = 3)
+	plot!(x, u, linestyle = :dash, label="Numerical solution", linewidth = 3)
+	#exact_sol
+end
+
+# ╔═╡ 116a4954-c6b9-474a-8ed4-627e02188b54
+md"""
+(``h=`` $(1/N))
+"""
+
+# ╔═╡ 52e5b8cc-d18a-4e20-bc66-54c010e2cc1f
+md"""
+The following plot shows the sparsity pattern of the matrix $A$ that represents the discretization with a finite difference method as a linear system.
+"""
+
+# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02
+begin
+	markersize = if N == 4
+	    15
+	elseif N == 5
+		11
+	elseif N == 6
+		7
+	elseif N == 7
+		4
+	elseif N == 8
+		3
+	else
+		2
+	end
+	markersizes_small = Dict(5=>11, 6=>7, 7=>4, 8=>3)
+	
+	spy(A + UniformScaling(5), markersize=markersize, markershape=:square)
+end
+
+# ╔═╡ 4376970c-ea81-415e-bdb8-c35d36751536
+md"""
+Below, you can also see the numerical values of the matrix A:
+"""
+
+# ╔═╡ 625a431f-9187-4c17-9411-3e9078499c38
+A
+
+# ╔═╡ 95fdd1b4-3e76-4dbd-976b-2243aea1e8b5
+md"""
+Note that for simplicity we use a classical or *dense* matrix representation here, where each entry of the matrix is stored explicitly. In a real application, one would use a *sparse* matrix representation that only stores the non-zero elements explicitly.
+"""
+
+# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda
+md"""
+## Auxiliary functions and internals
+If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions. Furthermore, implementation details that are required for this workbook to execute correctly are included here.
+"""
+
+# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa
+md"""
+### Load required packages
+This noteboook requires some additional packages to work:
+* Plots.jl is used for plotting the results
+* PlutoUI.jl is used to create the interactive GUI elements
+* LaTeXStrings.jl is used to make some mathematical notations look nicer
+* ShortCodes.jl allows to embed references to papers etc. more easily
+* `LinearAlgebra` is a Julia standard library module for linear algebra operations
+"""
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
+[deps]
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+version = "2.0.2+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.43+1"
+
+[[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 = "1.52.0+1"
+
+[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "1.4.1+1"
+"""
+
+# ╔═╡ Cell order:
+# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e
+# ╟─f87f000f-e0c9-46eb-a242-89a4290bd84f
+# ╠═f185aacd-0334-4217-8fa9-9fd0776c12c6
+# ╟─a7e57f8e-2032-419f-b957-12e90c3b9773
+# ╟─eb184f17-09c4-426e-80bb-f7d5867a01a4
+# ╟─25a2120b-e9cd-48bc-901e-5611a2330f99
+# ╟─116a4954-c6b9-474a-8ed4-627e02188b54
+# ╟─138b0481-7d42-4197-8ca2-b3fb073366a0
+# ╠═36f8657b-85db-4639-9941-6f7888a30b81
+# ╠═73c82ffa-a5e0-4040-8525-cc69a36a880d
+# ╠═336dd44e-1791-4aca-9258-ed958f704892
+# ╟─52e5b8cc-d18a-4e20-bc66-54c010e2cc1f
+# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02
+# ╟─4376970c-ea81-415e-bdb8-c35d36751536
+# ╟─625a431f-9187-4c17-9411-3e9078499c38
+# ╟─95fdd1b4-3e76-4dbd-976b-2243aea1e8b5
+# ╟─6b49c667-b16e-44f3-9f0d-2b8a62f5efda
+# ╟─b8f81a3f-d773-4e67-b8d7-e1734b04abaa
+# ╠═a22d543c-6a50-11ed-002d-7f2fe5bd7276
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002