Lecture 23: Solving Partial Differential Equations (PDEs) Numerically

Part II: Heat transport by ocean currents (two-dimensional advection and diffusion)

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1) Background: two-dimensional advection-diffusion

1.1) The two-dimensional advection-diffusion equation

Recall from Lecture 22 that the one-dimensional advection-diffusion equation is written as

$\frac{\partial T (x, t)}{\partial t} = - U \frac{\partial T}{\partial x} + κ \frac{\partial^{2} T}{\partial x^{2}},$

where $T (x, t)$ is the temperature, $U$ is a constant advective velocity and $κ$ is the diffusivity.

The two-dimensional advection diffusion equation simply adds advection and diffusion operators acting in a second dimensions $y$ (orthogonal to $x$ ).

$\frac{\partial T (x, y, t)}{\partial t} = u (x, y) \frac{\partial T}{\partial x} + v (x, y) \frac{\partial T}{\partial y} + κ (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}),$

where $\vec{u} (x, y) = (u, v) = u \hat{x} + v \hat{y}$ is a velocity vector field.

Throughout the rest of the Climate Modelling module, we will consider $x$ to be the longitundinal direction (positive from west to east) and $y$ to the be the latitudinal direction (positive from south to north).

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1.2) Multivariable shorthand notation

Conventionally, the two-dimensional advection-diffusion equation is written more succintly as

$\frac{\partial T (x, y, t)}{\partial t} = - \vec{u} \cdot \nabla T + κ \nabla^{2} T,$

using the following shorthand notation.

The gradient operator is defined as

$\nabla \equiv (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$

such that

$\nabla T = (\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y})$

and

$\vec{u} \cdot \nabla T = (u, v) \cdot (\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}) = u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} .$

The Laplacian operator $\nabla^{2}$ (sometimes denoted $Δ$ ) is defined as

$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$

such that

$\nabla^{2} T = \frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} .$

The divergence operator is defined as $\nabla \cdot []$ , such that

$\nabla \cdot \vec{u} = (\frac{\partial}{\partial x}, \frac{\partial}{\partial x}) \cdot (u, v) = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} .$

Note: Since seawater is largely incompressible, we can approximate ocean currents as a non-divergent flow, with $\nabla \cdot \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$ . Among other implications, this allows us to write:

\begin{align} \vec{u} \cdot \nabla T&= u\frac{\partial T(x,y,t)}{\partial x} + v\frac{\partial T(x,y,t)}{\partial y}\newline &= u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + T\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)\newline &= \left( u\frac{\partial T}{\partial x} + T\frac{\partial u}{\partial x} \right) + \left( v\frac{\partial T}{\partial y} + \frac{\partial v}{\partial y} \right) \newline &= \frac{\partial (uT)}{\partial x} + \frac{\partial (vT)}{\partial x}\newline &= \nabla \cdot (\vec{u}T) \end{align}

using the product rule (separately in both $x$ and $y$ ).

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1.3) The flux-form two-dimensional advection-diffusion equation

This lets us finally re-write the two-dimensional advection-diffusion equation as:

$\frac{\partial T}{\partial t} = - \nabla \cdot (\vec{u} T) + κ \nabla^{2} T$

which is the form we will use in our numerical algorithm below.

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2)

md"""

### 2)

"""

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begin

Δx = 0.04

Δy = 0.04

xs = (0. -Δx/2.:Δx:1. +Δx/2.)'

ys = (-1. -Δy/2.:Δy:1. +Δx/2.)

Nx = length(xs)

Ny = length(ys)

end;

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size(T)

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0.005

Δt = 0.005

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3×3 OffsetArray(::Matrix{Float64}, -1:1, -1:1) with eltype Float64 with indices -1:1×-1:1:
 0.0   1.0  0.0
 1.0  -4.0  1.0
 0.0   1.0  0.0

begin

diff_kernel = OffsetArray(zeros(Float64, 3,3), -1:1, -1:1)

diff_kernel[0, 0] = -4

diff_kernel[-1, 0] = 1.; diff_kernel[1, 0] = 1.;

diff_kernel[0, -1] = 1.; diff_kernel[0, 1] = 1.;

diff_kernel

end

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plot_kernel(diff_kernel)

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diffuse (generic function with 2 methods)

begin

function diffuse(T, j, i)

return κ.*sum(diff_kernel[-1:1,-1:1].*T[j-1:j+1, i-1:i+1])/(2Δx^2)

end

diffuse(T) = [diffuse(T, j, i) for j=2:Ny-1, i=2:Nx-1]

end

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3×3 OffsetArray(::Matrix{Float64}, -1:1, -1:1) with eltype Float64 with indices -1:1×-1:1:
  0.0  -1.0  0.0
 -1.0   0.0  1.0
  0.0   1.0  0.0

begin

adv_kernel = OffsetArray(zeros(Float64, 3,3), -1:1, -1:1)

adv_kernel[-1, 0] = -1.; adv_kernel[1, 0] = 1.;

adv_kernel[0, -1] = -1.; adv_kernel[0, 1] = 1.;

adv_kernel

end

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plot_kernel(adv_kernel)

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1×27 adjoint(::StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}) with eltype Float64:
 -0.02  0.02  0.06  0.1  0.14  0.18  0.22  …  0.78  0.82  0.86  0.9  0.94  0.98  1.02

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plot_kernel (generic function with 1 method)

plot_kernel(A) = heatmap(

collect(A),

color=:bluesreds, clims=(-maximum(abs.(A)), maximum(abs.(A))), colorbar=false,

xticks=false, yticks=false, size=(100, 100), xaxis=false, yaxis=false

)

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advect (generic function with 2 methods)

begin

function advect(T, j, i)

return .-(

sum(adv_kernel[0, -1:1].*(U[j, i-1:i+1].*T[j, i-1:i+1]))/(2Δx) .+

sum(adv_kernel[-1:1, 0].*(V[j-1:j+1, i].*T[j-1:j+1, i]))/(2Δy)

)

end

advect(T) = [advect(T, j, i) for j=2:Ny-1, i=2:Nx-1]

end

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update_ghostcells! (generic function with 1 method)

function update_ghostcells!(A; option="no-flux")

Atmp = @view A[:,:]

if option=="no-flux"

A[1, :] = Atmp[2, :]; Atmp[end, :] = Atmp[end-1, :]

A[:, 1] = Atmp[:, 2]; Atmp[:, end] = Atmp[:, end-1]

end

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50×25 Matrix{Float64}:
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  …  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
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 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  …  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
  ⋮                             ⋮          ⋱         ⋮                      
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  …  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0
 -0.0  -0.0  -0.0  -0.0  -0.0  -0.0  -0.0     -0.0  -0.0  -0.0  -0.0  -0.0  -0.0

advect(T)

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50×25 Matrix{Float64}:
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 ⋮                        ⋮              ⋱                 ⋮                   
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  …  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0     0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

diffuse(T)

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outgoing_thermal_radiation (generic function with 1 method)

function outgoing_thermal_radiation(T)

A .- B .* (T .- T0)

end

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plot(-20:20, α, ylim=(0,1))

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α (generic function with 1 method)

function α(T; α0=α0, αi=αi, ΔT=3.)

if T < -ΔT

return αi

elseif -ΔT <= T < ΔT

return αi + (α0-αi)*(T+ΔT)/(2ΔT)

elseif T >= ΔT

return α0

end

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52×27 Matrix{Float64}:
 45.4957   45.4957   45.4957   45.4957   …  45.4957   45.4957   45.4957   45.4957
 45.4957   45.4957   45.4957   45.4957      45.4957   45.4957   45.4957   45.4957
 45.4611   45.4611   45.4611   45.4611      45.4611   45.4611   45.4611   45.4611
 45.3921   45.3921   45.3921   45.3921      45.3921   45.3921   45.3921   45.3921
 45.2886   45.2886   45.2886   45.2886      45.2886   45.2886   45.2886   45.2886
 45.1508   45.1508   45.1508   45.1508   …  45.1508   45.1508   45.1508   45.1508
 44.9788   44.9788   44.9788   44.9788      44.9788   44.9788   44.9788   44.9788
  ⋮                                      ⋱                       ⋮        
 15.4315   15.4315   15.4315   15.4315      15.4315   15.4315   15.4315   15.4315
 14.3407   14.3407   14.3407   14.3407      14.3407   14.3407   14.3407   14.3407
 13.2461   13.2461   13.2461   13.2461      13.2461   13.2461   13.2461   13.2461
 12.1487   12.1487   12.1487   12.1487      12.1487   12.1487   12.1487   12.1487
 11.0498   11.0498   11.0498   11.0498   …  11.0498   11.0498   11.0498   11.0498
  9.95024   9.95024   9.95024   9.95024      9.95024   9.95024   9.95024   9.95024

Ss = [

S_peak .* (cos((y+1) * π/4) + .3)

for y in ys, x in xs[:]

]

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Error message

UndefVarError: S not defined

Stack trace

Here is what happened, the most recent locations are first:

from This cell: line 1
```
heatmap(S) |> as_png
```

heatmap(S) |> as_png

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---

absorbed_solar_radiation (generic function with 1 method)

function absorbed_solar_radiation(T)

absorption = 1.0 .- α.(T)

absorption .* Ss

end

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begin

# Initial conditions

T = [

for y in ys, x in xs[:]

]

t = Ref(0.)

end;

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0.0

T0 = 0.0

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0.377477

0.046875

begin

CFL_adv = maximum(V)*Δt/Δx

CFL_diff = κ*Δt/(Δx^2)

CFL_adv, CFL_diff

end

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for i in 1:10

timestep!(t, T)

end

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for i in 1:10

timestep2!(t, T)

end

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function timestep2!(t, T)

update_ghostcells!(T)

T[2:end-1, 2:end-1] .+= Δt.*(

advect(T) .+ diffuse(T) .+

(absorbed_solar_radiation(T)[2:end-1, 2:end-1]) .-

(outgoing_thermal_radiation(T)[2:end-1, 2:end-1])

)

t[] += Δt

end;

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Two-dimensional advection and diffusion

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Need boundary conditions still!

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plot(

-10:40, outgoing_thermal_radiation(-10:40),

xlabel="Temperature",

ylabel="Outgoing radiation"

)

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function timestep!(t, T)

update_ghostcells!(T)

T[2:end-1, 2:end-1] .+= Δt*(

advect(T) .+ diffuse(T) .+

(@view absorbed_solar_radiation(T)[2:end-1, 2:end-1]) .-

(@view outgoing_thermal_radiation(T)[2:end-1, 2:end-1])

)

t[] += Δt

end;

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speed:

@bind go Clock(.1)

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Increase global temperature by °C

Set global temperature to °C

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begin

temperature_control_event_handled

nT = 50

for i = 1:nT

timestep!(t, T)

end

plot_state()

end

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0.0

go; mean(T)

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0.015

κ = 0.015

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0.3

α0=0.3

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0.5

αi=0.5

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A = 11

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-0.7

B = -0.7

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35.0

S_peak = 35.0

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mean (generic function with 1 method)

mean(x) = sum(x) / length(x)

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plot_state (generic function with 1 method)

function plot_state()

X = repeat(xitp(xs), size(yitp(ys),1), 1)

Y = repeat(yitp(ys), 1, size(xitp(xs),2))

p = temperature_heatmap(T)

Nq = 4

quiver!(p, X[(Nq+1)÷2:Nq:end], Y[(Nq+1)÷2:Nq:end], quiver=(U[(Nq+1)÷2:Nq:end]./10., V[(Nq+1)÷2:Nq:end]./10.), color=:black, alpha=0.7)

plot!(p, xlims=(0., 1.), ylims=(-1.0, 1.0))

plot!(p, xlabel="longitudinal distance", ylabel="latitudinal distance")

plot!(p, clabel="Temperature")

as_png(p)

end

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Temperature control logic

begin

if !ismissing(temperature_control_event)

e = temperature_control_event

if e["type"] == "add"

T .+= e["value"]

elseif e["type"] == "set"

T .= e["value"]

end

temperature_control_event_handled = true

Text("Temperature control logic")

end

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heatmap(xs', ys, ψ̂)

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temperature_heatmap (generic function with 1 method)

function temperature_heatmap(T)

levels = -10:1.0:40

p = contourf(xs', ys, T,

levels=levels,

color=:bluesreds, colorbar_title="Temperature [°C]",

lw=0,

clims=extrema(levels)

)

contour!(p, xs', ys, T,

levels=[0],

color=:white,

lw=3

)

end

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Setting up the velocity field

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begin

ϵ = 0.05

xψ = (-Δx:Δx:1. +Δx)'

yψ = (-1-Δy:Δy:1. +Δy)

# See page 595 of Vallis Edt.2

ψ̂(x,y) = π*sin.(π*y) * (

1 .- x - exp.(-x/(2*ϵ)) .* (

cos.(√3*x/(2*ϵ)) .+

(1. /√3)*sin.(√3*x/(2*ϵ))

)

.+ ϵ*exp.((x .- 1.)/ϵ)

)

u,v = diagnose_velocities(ψ̂(xψ, yψ))

U = xitp(u) ./10.

V = yitp(v) ./10.

U[1,:] .= 0.; V[1,:] .= 0.;

U[end,:] .= 0.; V[end,:] .= 0.;

U[:,1] .= 0.; V[:,1] .= 0.;

U[:,end] .= 0.; V[:,end] .= 0.;

end;

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diagnose_velocities (generic function with 1 method)

begin

∂x(ϕ) = (ϕ[:,2:end] - ϕ[:,1:end-1])/Δx

∂y(ϕ) = (ϕ[2:end,:] - ϕ[1:end-1,:])/Δy

xpad(ϕ) = hcat(zeros(size(ϕ,1)), ϕ, zeros(size(ϕ,1)))

ypad(ϕ) = vcat(zeros(size(ϕ,2))', ϕ, zeros(size(ϕ,2))')

xitp(ϕ) = 0.5*(ϕ[:,2:end]+ϕ[:,1:end-1])

yitp(ϕ) = 0.5*(ϕ[2:end,:]+ϕ[1:end-1,:])

function diagnose_velocities(ψ)

u = ∂y(ψ)

v = -∂x(ψ)

return u,v

end

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begin

import Pkg

Pkg.activate(mktempdir())

Pkg.add([

"Plots",

"PlutoUI",

"Images",

"FileIO",

"ImageMagick",

"ImageIO",

"OffsetArrays",

"ThreadsX",

"Strided",

])

using Plots

using PlutoUI

using Images

using OffsetArrays

using ThreadsX

using Strided

end

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  Activating new project at `/tmp/jl_tbPDGm`
    Updating registry at `~/.julia/registries/General.toml`
   Resolving package versions...
   Installed CUDA_Driver_jll ──────── v0.5.0+1
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   Installed CUDA_Runtime_jll ─────── v0.6.0+0
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   Installed GPUArrays ────────────── v8.8.1
   Installed LLVM ─────────────────── v6.1.0
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   Installed StridedViews ─────────── v0.3.2
   Installed Strided ──────────────── v2.2.0
   Installed KernelAbstractions ───── v0.9.34
   Installed PackageExtensionCompat ─ v1.0.2
   Installed Random123 ────────────── v1.6.2
   Installed UnsafeAtomicsLLVM ────── v0.1.5
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   Installed UnsafeAtomics ────────── v0.2.1
   Installed LLVMExtra_jll ────────── v0.0.23+0
   Installed GPUCompiler ──────────── v0.21.4
   Installed CUDA ─────────────────── v4.4.2
    Updating `/tmp/jl_tbPDGm/Project.toml`
  [5789e2e9] + FileIO v1.17.0
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  [916415d5] + Images v0.26.2
  [6fe1bfb0] + OffsetArrays v1.15.0
  [91a5bcdd] + Plots v1.40.7
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Frontmatter

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Lecture 23: Solving Partial Differential Equations (PDEs) Numerically

1) Background: two-dimensional advection-diffusion

1.1) The two-dimensional advection-diffusion equation

1.2) Multivariable shorthand notation

1.3) The flux-form two-dimensional advection-diffusion equation

2)

Two-dimensional advection and diffusion

Need boundary conditions still!

Setting up the velocity field