Lecture04Explicit.jl

### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# ╔═╡ 254b7584-7a7e-43f8-a17d-d94b121b1416
begin
using Plots
using LinearAlgebra
end

# ╔═╡ 46ca86db-be88-4dc5-aa03-7b60d1e54f6a
md"""
# Simple ODEs
Ideally, a numerical method for solving ordinary differential equations

$$\dot x(t)=f(t,x(t)), \quad x(t_0)=x_0,$$

should be able to solve exactly those equations often posed in high-school textbooks.

Examples include:
### Zero
$$1) \quad \dot x(t)=0, \quad x(t_{0}) = x_0\\$$
### Constant
$$2) \quad \dot x(t) = \lambda, \quad x(t_0)=x_0$$
### Linear
$$3) \quad \dot x(t) =\lambda x(t), \quad x(t_0)=x_0$$
"""

# ╔═╡ 7f187f3f-0afd-401e-a384-d0ab1551e7e1
md"""
## Their solutions
If you are taking this course, it is good to know that their solutions are:
### Zero
$$1) \quad x(t)=x_0\\$$
### Constant
$$2) \quad x(t) = x_{0}+\lambda(t-t_0)$$
### Linear
$$3) \quad x(t) =x_0\exp(\lambda(t-t_0))$$
"""

# ╔═╡ 770f43c0-2768-40a1-bb8a-384601a0d0a8
md"""
## Explicit Euler attempt
Utilizing perhaps the simplest scheme

$$x_{k+1}=x_{k}+hf(x_{k}),$$

with fixed step-size $h$, we find the following:
### Zero
$$x_{k+1}=x_{k}=\dots=x_0$$
**Exact!**
### Constant
$$x_{k+1}=x_{k}+h\lambda =(x_{k-1}+h\lambda)+h\lambda=\dots=x_{0}+(k+1)h\lambda$$
**Exact!**
### Linear
$$x_{k+1}=x_{k}+h\lambda x_{k}=(1+h\lambda)x_{k}=\dots=(1+h\lambda)^{k+1}x_{0}$$
**Approximation!**
From the definition of the exponential function,
$$\exp(x)=\lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}$$
we find that an explicit Euler scheme approximates the term

$$\exp(\lambda(t_{k+1}-t_0))=\exp((k+1)h\lambda)=\lim_{n\to\infty}\left(1+\frac{(k+1)h\lambda}{n}\right)^{n},$$
by halting the limit sequence at $n=k+1$.
"""

# ╔═╡ f8863d59-0f81-4833-9d2f-f3a021739d26
md"""
# Accuracy
Although we as humans are error-prone, we are able to derive and calculate the analytical solution to all of the above ODEs. However, as we have now seen theoretically, a numerical method may be limited in producing accurate answers.
"""

# ╔═╡ 40c76196-753c-48f4-a310-564df338dbd5
md"""
## Truncation error
Sometimes also referred to as discretization error. An error inherent to the numerical method due to mathematical approximations.
### Example
The explicit Euler scheme can be derived by approximating the integral solution to an ODE using a simple left-hand Riemann quadrature:

$$x(t_{0}+h)=x(t_{0})+\int_{t_{0}}^{t_{0}+h}f(t,x(t))\,\mathrm{d}t\approx x(t_{0})+hf(t_0,x(t_0))$$

### Definitions
For numerical methods for ODEs we discern two error definitions:

#### Local truncation error
The local truncation error $\tau_{k}$ at step $k$ is defined as the error made by a single step of a numerical integration scheme

$$\tau_{k+1}=x_{k+1}-x_{k}(t_{k+1}),$$

where $x_k(t_{k+1})$ is defined as the analytical solution of the ODE
$$\dot x_k(t)=f(t,x_k(t)), \quad x_k(t_{k})=x_k$$

**Drawing?**

#### Global truncation error
The global truncation error $e_{k}$ at step $k$ is defined as the difference between the numerical and analytical solutions

$$e_{k+1}=x_{k+1}-x(t_{k+1})$$

**Drawing?**

#### Method order
If the local truncation error is of **asymptotic** order

$$\tau_{k+1}=O(h^{p+1}), \quad h\to0$$

for a given exponent $p$, then the numerical scheme is said to have **method order** $p$.

**What about the order of the global truncation error?** Requires some math..
"""

# ╔═╡ e78d20c8-7ef6-4fb7-855a-6db631f99f9b
md"""
### Analyzing local truncation error
Let us consider the local truncation error of the explicit Euler scheme. With help from a first-order Taylor expansion around the previous time point, we see that for $h \to 0$

$$\begin{align*}
\tau_{k+1}&=x_{k+1}-x_{k}(t_{k+1})\\
&=x_k+hf(t_k,x_{k})-\left[x_k(t_k)+hf(t_{k,}x_{k}(t_{k}))+O(h^2)\right]\\
&=x_k+hf(t_{k},x_{k})-\left[x_k+hf(t_{k},x_{k})+O(h^2)\right]\\
&=O(h^{2})
\end{align*}$$

Therefore, we find that the explicit Euler scheme has **method order** $p=1$.

**In this case, halving the step-size will quarter the error!**
"""

# ╔═╡ 51564216-03f1-4f7f-98da-5e6e47d10a77
md"""
## Round-off error
If we were to implement the explicit Euler scheme on a computer, we might expect that it should still find exact solutions to both the zero and constant ODEs above. However, this can not be expected!

Another, sometimes forgotten, source of error is numerical round-off. An unfortunate limitation when working with finite precision floating points numbers.

Problems occur due since computer representations of the reals are limited by
1) Magnitude ~ $[10^{-308},10^{308}]$
2) Precision ~ $10^{-16}$
**For IEEE 754 double-precision binary floating-point format**
### Beware
#### Catastrophic cancellation
When subtracting two nearly equal numbers, the result will suffer a loss in number of significant digits!
#### Cumulative round-off
Cumulation of many small round-off errors may be significant!
"""

# ╔═╡ 5c99d554-6741-4c0e-9783-771680e414fe
md"""
# Stability
So far, we have seen that the step-size has to be chosen in order to control the local truncation error. However, the step-size is also directly related to the stability of our numerical method.

Note, that there is a difference between the stability of the analytical solutions and the numerical solutions.

Stability theory for the solutions of ODEs is well-developed  and interesting theoretical topic. However, once it has been determined that we may expect stable analytical solutions, we should be interested in if the same holds for our numerical approximations.
"""

# ╔═╡ 5c226710-54fc-4bee-8e71-5a0d0061c0f0
md"""
## A curious example
We might think that so far, everything has gone well, so what could go wrong?

A surprising phenomenon appears in certain types of equations. Consider the following example

$$\dot x(t) = -50(x(t)-\cos(t)), \quad x(0)=x_{0}.$$

Plotting its solution curves, we find:
"""

# ╔═╡ eee070f2-f671-4217-9a39-38d492a4a7a7
begin
	function analytical(t, a)
		out = 2501.0 * a * exp(-50.0 * t)
		out -= 2500.0 * exp(-50.0 * t)
		out += 50.0 * sin(t)
		out += 2500.0 * cos(t)
		out /= 2501
		return out
	end

	_a = 10.0 .^ collect(0:1:50)
	_a = [-_a;_a]
	_t = collect(0.0:0.01:1.5)
	_s = [analytical(t, a) for t in _t, a in _a]

	plot(_t, _s, label="", ylim=(0,1.5))

end

# ╔═╡ 1c5e654d-4699-4a00-9ba5-f5da72969d35
md"""
It is clear to see that all solution curves have a rapid transient phase towards the equilibrium solution $x(t) \approx \cos(t)$.

Imagine that we were to solve this equation using the explicit Euler scheme. If, by chance, we choose an initial condition $x_{0}\neq 1$, then the numerical solution curve would produce a "bumpy ride".
"""

# ╔═╡ 63489e4c-624f-4f03-8045-6b94e4876b10
md"""
## Stiffness
This phenomenon is called **stiffness** and its definition is not universally agreed upon.

Stiff equations are:
1) Problems for which explicit methods don't work
2) Problems for which certain numerical methods are numerically unstable
3) Problems for which the step-size has to be taken extremely small

Stiffness appears when the solution curves are "very stable".
"""

# ╔═╡ 48813f57-3029-4c91-814e-3603954d4c00
md"""
## Stability analysis
To grasp whether a numerical method may suffer from instability, we perform a stability analysis.
### Linear stability theory
Given the ODE

$$\dot x(t)=f(t,x(t)), \quad x(t_0)=x_0,$$

classical stability theory for the analytical solutions tell us that we have to investigate the stability of solutions to

$$\dot X(t)=JX(t), \quad X(t_0)=0,$$

where $J=\partial_{x}f(t_{0},x(t_{0}))$ is the Jacobian at some steady-state $f(t_{0},x(t_0))=0$.


In this case, the linear stability theory tells us that the steady-state solution of the original equation is stable if all eigenvalues of the Jacobian have negative real part:

$$\text{Re}(\lambda)<0, \quad \forall \lambda \in \lambda(J).$$

"""

# ╔═╡ 05882b94-54ee-4dbb-876c-02a7ec5bbe49
md"""
### Test equation
A simplified version of the above "test" equation (with a non-trivial solution) is known as the **Dahlquist test equation**

$$\dot x(t)=\lambda x(t), \quad x(0)=1,$$

with parameter $\lambda \in \mathbb{C}$. It is the primary benchmark for numerical methods. From before, we know its analytical solution:

$$x(t)=\exp(\lambda t)$$

We also know explicit Euler **approximates** the solution!

"""

# ╔═╡ cc2977bc-145c-4ed9-a1fc-d157f7c761b4
md"""
#### Stability analysis for explicit Euler
Applied to the test equation, we recall that the explicit Euler scheme becomes a linear recurrence relation

$$x_{k+1}=x_{k}+h\lambda x_k=(1+h\lambda)x_k$$

If we want the numerical solution to remain stable, we must require that

$$|1+h\lambda| \leq 1$$

Hence, for a given problem, where $\lambda \in \mathbb{C}$ is fixed, this puts an explicit condition on step-sizes $h$ which guarantee stability. The corresponding **stability region** is shown below:
"""

# ╔═╡ 6dd7b689-9447-4804-be86-b4a38034f915
begin
	function explicit_stability(z)
		out = 1+z
	end

	_r = -2.5:0.01:1.5
	_i = -1.75:0.01:1.75
	_R = [explicit_stability(r+i*im) for r in _r, i in _i]
	_R = abs.(_R)
	heatmap(_r, _i, _R', c=:RdBu, xlabel="Re(hλ)", ylabel="Im(hλ)", title="Explicit Euler stability")
	vline!([0.0], color=:black, lw=2, label="")
	hline!([0.0], color=:black, lw=2, label="")
	plot!(cos.(0:0.1:2pi).-1,sin.(0:0.1:2pi), color=:black, lw=2, label="")
end

# ╔═╡ dd2c6725-dfd4-4f5f-ad3b-46728fe36568
md"""
#### Stability analysis for implicit Euler
A similar analysis is also performed for the implicit Euler scheme

$$x_{k+1}=x_{k}+hf(t_{k},x_{k})$$

Here we see for the test equation that

$$x_{k+1}=x_{k}+h\lambda x_{k+1} =\frac{1}{1-h\lambda}x_k$$

Again, stability requires that

$$\left|\frac{1}{1-h\lambda}\right|\leq1$$

The corresponding stability region is shown below:
"""

# ╔═╡ 174c6695-89b3-4786-8289-e68be0d42e1f
begin
	function implicit_stability(z)
		out = 1/(1-z)
	end

	_ri = -1.5:0.01:2.5
	_ii = -1.75:0.01:1.75
	_Ri = [implicit_stability(r+i*im) for r in _ri, i in _ii]
	_Ri = abs.(_Ri)
	heatmap(_ri, _ii, _Ri', c=:RdBu, clim=(0,3.05), xlabel="Re(hλ)", ylabel="Im(hλ)", title="Implicit Euler stability")
	vline!([0.0], color=:black, lw=2, label="")
	hline!([0.0], color=:black, lw=2, label="")
	plot!(cos.(0:0.1:2pi).+1,sin.(0:0.1:2pi), color=:black, lw=2, label="")
end

# ╔═╡ 7dd538c9-6fb0-4736-966a-0c43349a7638
md"""
**How would we then expect the implicit Euler scheme handle the above stiff equation?**
"""

# ╔═╡ a634c6c7-f387-4720-afc1-d22ce9f7ce83
md"""
#### General stability analysis
In general, if we can write our numerical method applied to the test equation as a recurrence relation

$$x_{k+1}=R(h\lambda)x_{k}$$
then the stability region is given by all $h\in \mathbb{R}$ and $\lambda \in \mathbb{C}$ satisfying

$$|R(h\lambda)|\leq 1$$

**Note the difference between the continuous linear stability condition and discrete linear stability condition.**
"""

# ╔═╡ c9ec776e-e445-443d-b8e0-ce1cd02cb5da
md"""
# Break
**5-15 minutes?**
"""

# ╔═╡ 073e5c9c-a3d7-41bb-95d0-4c3fb9c750e3
md"""
# Error estimation
We now know that it is crucial to choose the step-size correctly, in order to enjoy both accuracy and stability. However, for accuracy, the step-size was related to the local error of our numerical method, which is unknown.

**Let us estimate it!**
"""

# ╔═╡ 08c6c980-9997-47a5-b591-628c8746a6ff
md"""
## Step-doubling
We know already that if we half the step-size, then asymptotically we will quarter the local error. Therefore, we can compare the numerical solution obtained by a single step, to that obtained by taking two half steps.

We perform a rudimentary theoretical analysis for scalar equations.

**See Lecture04C slides for the analysis on the test equation.**
"""

# ╔═╡ ae73a126-6705-45ee-8637-665a84e9f1c3
md"""
### Single step
Given a method of order $p$, we know that, asymptotically, for one step from $t_{k}$ to $t_{k+1}$ we have a local truncation error of the order

$$\tau_{k+1} = x_{k+1}-x_k(t_{k+1})=Ch^{p+1}+O(h^{p+2})$$

### Double step
Using the same method, we first take a half step with local truncation error

$$\hat{\tau}_{k+\frac{1}{2}}=\hat{x}_{k+\frac{1}{2}}-x_k(t_{k+\frac{1}{2}})=C\left(\left(h/2\right)^{p+1}\right)+O(h^{p+2})$$

and a second half step, which will have a similar local truncation error

$$\hat\tau_{k+1}=\hat{x}_{k+1}-x_{k+\frac{1}{2}}(t_{k+\frac{1}{2}})=C\left(\left(h/2\right)^{p+1}\right)+O(h^{p+2})$$

Using Taylor expansions, it is possible to show that

$$\hat{x}_{k+1}-x_{k}(t_{k+1})=2C\left(\left(h/2\right)^{p+1}\right)+O(h^{p+2})$$

"""

# ╔═╡ 56809f6e-6caf-4f4a-8e0b-fbc75633a04d
md"""
### Error estimate
Consequently, we find that that the error estimate asymptotically

$$\begin{align*}
\hat\tau_{k+1}=\hat{x}_{k+1}-x_{k+1} &= 2C\left(\left(h/2\right)^{p+1}\right)-Ch^{p+1}+O(h^{p+2})\\
&= KCh^{p+1}+O(h^{p+2}), \quad K\geq1
\end{align*}$$

We now see that this estimate of the local truncation error asymptotically follows a $K$ multiple of the original local truncation error

$$\hat\tau_{k+1} =K\tau_{k+1} \geq \tau_{k+1},$$

and therefore, we can use it to overestimate the error!
"""

# ╔═╡ 5e720150-4cf5-40b1-91e9-b0aaaae5ab71
md"""
#### Richardson Extrapolation
In a method known as Richardson extrapolation, the constant $K$ can be used to describe a method with one order higher $(p+1)$. Note that it is straightforward to compute the constant $K$.
"""

# ╔═╡ ac75df14-444b-48b5-97e8-87b781be101b
md"""
# Adaptive step-size control
## Accepting steps
Given the estimate of our local truncation error $\hat{\tau}_{k+1}$, we are now able to judge, whether or not our current step was computed accurately enough.

Setting an absolute tolerance $\varepsilon_\text{abs}>0$, we check whether the absolute truncation error is small enough

$$\|\hat\tau_{k+1}\|\leq \varepsilon_\text{abs}$$

If we further want control over the relative error, we set a relative tolerance $\varepsilon_\text{rel}$ and check whether

$$\|\hat\tau_{k+1}\|\leq \|[\varepsilon_\text{abs},\varepsilon_\text{rel}\|x_{k+1}\|]^T\|$$

Here, the choice of "norm" $||\cdot||$ is free to choose.

Often, we will consider scaled error estimates and check

$$\|r_{k+1}\|\leq 1$$

and to be extra sure, we sometimes include a safety factor $\alpha < 1$ (often $\alpha=0.8,0.95$)

$$\|r_{k+1}\|\leq \alpha$$
"""

# ╔═╡ 56d515b4-17db-405b-a046-42477e606020
md"""
## Adjusting steps
Assuming that the current error estimate was found too large with the step-size $h_{k}$, we consult the asymptotic error

$$\|r_{k+1}\|=\hat{C}h_{k}^{p+1}\leq\alpha$$

Rearranging the inequality and taking the largest step yields

$$\hat h_{k}=\left(\frac{\alpha}{\hat{C}}\right)^{\frac{1}{p+1}}$$

Rearranging the left-hand side equality gives us an estimate for the constant

$$\hat{C}=\|r_{k+1}\|h_{k}^{-(p+1)},$$

which yields the final choice of our new step-size

$$\hat{h}_{k}=h_{k}\left(\frac{\alpha}{\|r_{k+1}\|}\right)^{\frac{1}{p+1}}$$

This is known as **asymptotic step-size control**!

**Remark**: In case the step was accepted, the asymptotic step-size controller also gives a method to adjust the step-size in order to asymptotically hit the local truncation error tolerance.
"""

# ╔═╡ 3d709ffe-d04d-41e4-9ddb-c543d74ad539
md"""
## Initial step-size
A heuristic is needed to choose the initial step-size.

**See Lecture04B slides on Learn!**
"""

# ╔═╡ 069d49de-b8f7-469f-9bc6-445ff3cc680d
md"""
# Numerical experiments
The above considerations have been implemented below.

Here we have a method to perform adaptive integration with an explicit Euler scheme, as well as the classical Runge-Kutta scheme.

Let us now test how our routines behave with respect to
- Accuracy
- Stability
- Efficiency
"""

# ╔═╡ 654c5fc7-dda5-4d55-a8fc-f6702ebc7c11
begin
	function adaptive_integrator(f, tspan, x₀, p; h₀=1e-6, reltol=1e-3, abstol=1e-6, method=euler_step!)
        # Error controller parameters
        safety_factor = 0.8
        min_factor = 0.1
        max_factor = 10.0
        order = if (method == euler_step!) 1.0 else 4.0 end

        # Integration interval
        t₀, tₛ = tspan

        # Initial condition
        t = t₀
        t₊ = t₀ + h₀
        h = h₀
        x = deepcopy(x₀)

        # Cache
        x₊ = zero(x)
        x₊₊ = zero(x)
        x₊₊₊ = zero(x)
        dx = zero(x)
        e = zero(x)

		# Counters
		nfun = 0
		naccept = 0
		nreject = 0

        # Output
        ts = [t]
        xs = [deepcopy(x)]
        rs = [0.0]

        # Main loop
        while t < tₛ
            if t₊ > tₛ
                t₊ = tₛ
                h = t₊ - t
            end
            # Compute full step
            method(x₊, f, dx, x, p, t, h)

            # Compute two half steps
            method(x₊₊, f, dx, x, p, t, 0.5h)
			method(x₊₊₊, f, dx, x₊₊, p, t + 0.5h, 0.5h)

			if (method == euler_step!)
				nfun += 3
			else
				nfun += 12
			end

            # Get error estimate
            e .= x₊₊₊ .- x₊
            e ./= abstol .+ reltol .* max.(abs.(x₊₊₊), abs.(x₊))
            r = norm(e)

            if r ≤ 1
                # Accept the step
				naccept += 1
                t += h
                x .= x₊₊₊
                push!(ts, t)
                push!(xs, deepcopy(x))
                push!(rs, r)
			else
				nreject += 1
			end

            # Asymptotic step-size control
            h *= max(min_factor, min(max_factor, (safety_factor / r)^(1/(order+1))))
        end

		return ts, hcat(xs...), rs, nfun, naccept, nreject
	end

	function euler_step!(x₊, f, dx, x, p, t, h)
        # Explicit Euler scheme
        f(dx, x, t, p)
        x₊ .= x .+ h .* dx

        return nothing
    end

    function rk4_step!(x₊, f, dx, x, p, t, h)
        #Non-optimized classical Runge-Kutta scheme
        f(dx, x, t, p)
        k₁ = h .* dx
        f(dx, x .+ 0.5 .* k₁, t + 0.5 .* h, p)
        k₂ = h .* dx
        f(dx, x .+ 0.5 .* k₂, t + 0.5 .* h, p)
        k₃ = h .* dx
        f(dx, x .+ k₃, t + h, p)
        k₄ = h .* dx
        x₊ .= x .+ (k₁ .+ 2 .* k₂ .+ 2 .* k₃ .+ k₄) ./ 6
        return nothing
    end

end

# ╔═╡ 3ec61edc-f080-4c60-8583-c6e467628d69
md"""
## Benchmark
Following the modeling slides presented in the previous lectures, we have implemented a dispersive plug-flow reactor to benchmark on.

The dispersive PFR is a good test case, since its parameters can induce stiffness.

**Consult the Courant-Friedrichs-Lewy stability condition and Von Neumann stability analysis**
"""

# ╔═╡ a1ef93a5-710e-4051-8190-0279c93bbb73
function f(dx, x, t, p)
    # States and boundary conditions
    dcA = dx
    cA = x
    cAᵢₙ = p[5]

    # Mesh
    n = size(cA, 1)
    h = p[1]

    # Parameters
    v = p[2]
    D = p[3]
    k = p[4]

    # Flux
    N = zeros(n + 1)

    # Convective flux (upwinding)
    if v > 0
        N[1] = v * cAᵢₙ
        N[2:end] .= v .* cA
    else
        N[1:end-1] .= v .* cA
        N[end] = v * cAᵢₙ
    end

    # Diffusive flux
    N[2:end-1] .-= (D / h) .* (cA[2:end] .- cA[1:end-1])

    # Reaction rate
    dcA .= -2.0 .* k .* cA .* cA

    # Mass balances
    dcA  .-= (N[2:end] .- N[1:end-1]) ./ h

    return nothing
end

# ╔═╡ cb764f34-c959-4936-8902-7da7ebadf439
md"""
### Problem definition
"""

# ╔═╡ 0dacd922-fdf9-4f80-b9e7-f12a68720d79
begin
	## Problem
	# Parameters
	v = 1.0
	D = 2e-1
	k = 0.5

	# Mesh
	n = 20
	L = 1.0
	h = L / n
	zs = collect(range(0.0, L, length=n))

	## Initialization
	tₛ = 10.0
	tspan = (0.0, tₛ)
	c₀ = 0.1 .* ones(n)
	cᵢₙ = 0.2
	p = [h, v, D, k, cᵢₙ]

	## Integrate
	ts, xs, rs, nfun, naccept, nreject = adaptive_integrator(
		f, tspan, c₀, p;
		reltol=1e-3, abstol=1e-6,
		#method=euler_step!,
		method=rk4_step!,
	)

	nothing
end

# ╔═╡ d17aa551-9b9e-48a2-a699-24c2371fdb85
plot(ts, xs'[:, end], label="", ylabel="Concentration cA", xlabel="Time t", title="Outlet", lw=2)

# ╔═╡ cb6a3f3a-6b9c-4944-8ab0-870ee754ca35
begin
	plot(zs, xs[:,1:10:end], label="", ylabel="Concentration cA", xlabel="Length z", title="Profiles", lw=2)
end

# ╔═╡ 58d369b4-c52b-4af9-be3a-955edd253bf6
surface(ts, zs, xs, title="Concentration surface", xflip = true, camera = (75,35))

# ╔═╡ 00e1d80f-893e-47dd-8592-faeb71cb4744
md"""
## Accuracy & stability
"""

# ╔═╡ 235031fa-9d28-4398-becc-559eb59c4bd5
begin
	errorplot=plot(ts[2:end], rs[2:end], label="||r||", xlabel="Time t", ylabel="Error norm ||r||", lw=2, title="Error")
	hline!([1.0], label="Accept step", lw=2)
	hline!([0.8], label="Error goal", lw=2)

	hplot=plot(ts[2:end], diff(ts), label="h", xlabel="Time t", ylabel="Step-size h", lw=2, title="Step-size")

	plot(errorplot, hplot, layout=(2,1))
end

# ╔═╡ 03bcdf8a-6ef4-4056-a411-937e89c40789
begin
	@info "Function evaluations: $(nfun)"
	@info "Accepted steps: $(naccept)"
	@info "Rejected steps: $(nreject)"
end

# ╔═╡ 5bac1c45-1b54-455a-9150-25e5d65c9008
md"""
**To test the accuracy and stability, try to modify:**
- The diffusion parameter `D` and reaction parameter `k`
	- Which parameter effects the problem stiffness?
- The method `euler_step!` vs. `rk4_step!`
	- Which integrator performs the best?
	- At which tolerance?
	- At which "level" of stiffness?
"""

# ╔═╡ 5b205537-307a-4f1e-9739-62b1e1929467
md"""
## Efficiency
Let us reality-check our initial numerical tests. We benchmark both methods based on their individual **work-precision diagram**.
"""

# ╔═╡ d84473d0-f94d-411e-950b-f73ebb5cce37
function work_precision(f, tspan, x₀, p)
        # Sequence of equal reltol=abstol=tol in log-scape
        tols = 10.0 .^ collect(range(-8, -2, length=10))

        nfuns = zeros(length(tols), 2)

        for method in [euler_step!, rk4_step!]
            for (i, tol) in enumerate(tols)
                _, _, _, nfun, _, _ = adaptive_integrator(
                    f, tspan, x₀, p;
                    reltol=tol, abstol=tol,
                    method=method,
                )
                nfuns[i, method == euler_step! ? 1 : 2] = nfun
            end
        end

        return tols, nfuns
    end

# ╔═╡ 1b2b8427-c417-4d72-998b-ecf6f3687a15
begin
	#tols, nfuns = work_precision(f, tspan, c₀, p)
	#scatter(tols, nfuns, xscale=:log10, yscale=:log10, xlabel="Tolerance εₐ=εᵣ=ε", #ylabel="Function evaluations", markersize=5, label=["Explicit Euler" "Classical #RK4"])
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

[compat]
Plots = "~1.40.9"
"""

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version = "1.3.243+0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "85c7811eddec9e7f22615371c3cc81a504c508ee"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+2"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "5db3e9d307d32baba7067b13fc7b5aa6edd4a19a"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.36.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "a2fccc6559132927d4c5dc183e3e01048c6dcbd6"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.13.5+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "7d1671acbe47ac88e981868a078bd6b4e27c5191"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.42+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "beef98d5aad604d9e7d60b2ece5181f7888e2fd6"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.6.4+0"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "326b4fea307b0b39892b3e85fa451692eda8d46c"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.1+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "3796722887072218eabafb494a13c963209754ce"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "9dafcee1d24c4f024e7edc92603cedba72118283"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+3"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e9216fdcd8514b7072b43653874fd688e4c6c003"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.12+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "807c226eaf3651e7b2c468f687ac788291f9a89b"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.3+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "89799ae67c17caa5b3b5a19b8469eeee474377db"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.5+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "d7155fea91a4123ef59f42c4afb5ab3b4ca95058"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.6+3"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "6fcc21d5aea1a0b7cce6cab3e62246abd1949b86"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.0+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "984b313b049c89739075b8e2a94407076de17449"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.2+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "a1a7eaf6c3b5b05cb903e35e8372049b107ac729"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.5+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "b6f664b7b2f6a39689d822a6300b14df4668f0f4"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.4+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "a490c6212a0e90d2d55111ac956f7c4fa9c277a6"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+1"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c57201109a9e4c0585b208bb408bc41d205ac4e9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.2+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "1a74296303b6524a0472a8cb12d3d87a78eb3612"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "dbc53e4cf7701c6c7047c51e17d6e64df55dca94"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+1"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[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", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "ab2221d309eda71020cdda67a973aa582aa85d69"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+1"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6dba04dbfb72ae3ebe5418ba33d087ba8aa8cb00"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.1+0"

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "622cf78670d067c738667aaa96c553430b65e269"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+0"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6e50f145003024df4f5cb96c7fce79466741d601"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.56.3+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0ba42241cb6809f1a278d0bcb976e0483c3f1f2d"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+1"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "522c1df09d05a71785765d19c9524661234738e9"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.11.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "d7b5bbf1efbafb5eca466700949625e07533aff2"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.45+1"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[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 = "63406453ed9b33a0df95d570816d5366c92b7809"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+2"
"""

# ╔═╡ Cell order:
# ╟─254b7584-7a7e-43f8-a17d-d94b121b1416
# ╟─46ca86db-be88-4dc5-aa03-7b60d1e54f6a
# ╟─7f187f3f-0afd-401e-a384-d0ab1551e7e1
# ╟─770f43c0-2768-40a1-bb8a-384601a0d0a8
# ╟─f8863d59-0f81-4833-9d2f-f3a021739d26
# ╟─40c76196-753c-48f4-a310-564df338dbd5
# ╟─e78d20c8-7ef6-4fb7-855a-6db631f99f9b
# ╟─51564216-03f1-4f7f-98da-5e6e47d10a77
# ╟─5c99d554-6741-4c0e-9783-771680e414fe
# ╟─5c226710-54fc-4bee-8e71-5a0d0061c0f0
# ╟─eee070f2-f671-4217-9a39-38d492a4a7a7
# ╟─1c5e654d-4699-4a00-9ba5-f5da72969d35
# ╟─63489e4c-624f-4f03-8045-6b94e4876b10
# ╟─48813f57-3029-4c91-814e-3603954d4c00
# ╟─05882b94-54ee-4dbb-876c-02a7ec5bbe49
# ╟─cc2977bc-145c-4ed9-a1fc-d157f7c761b4
# ╟─6dd7b689-9447-4804-be86-b4a38034f915
# ╟─dd2c6725-dfd4-4f5f-ad3b-46728fe36568
# ╟─174c6695-89b3-4786-8289-e68be0d42e1f
# ╟─7dd538c9-6fb0-4736-966a-0c43349a7638
# ╟─a634c6c7-f387-4720-afc1-d22ce9f7ce83
# ╟─c9ec776e-e445-443d-b8e0-ce1cd02cb5da
# ╟─073e5c9c-a3d7-41bb-95d0-4c3fb9c750e3
# ╟─08c6c980-9997-47a5-b591-628c8746a6ff
# ╟─ae73a126-6705-45ee-8637-665a84e9f1c3
# ╟─56809f6e-6caf-4f4a-8e0b-fbc75633a04d
# ╟─5e720150-4cf5-40b1-91e9-b0aaaae5ab71
# ╟─ac75df14-444b-48b5-97e8-87b781be101b
# ╟─56d515b4-17db-405b-a046-42477e606020
# ╟─3d709ffe-d04d-41e4-9ddb-c543d74ad539
# ╟─069d49de-b8f7-469f-9bc6-445ff3cc680d
# ╟─654c5fc7-dda5-4d55-a8fc-f6702ebc7c11
# ╟─3ec61edc-f080-4c60-8583-c6e467628d69
# ╟─a1ef93a5-710e-4051-8190-0279c93bbb73
# ╟─cb764f34-c959-4936-8902-7da7ebadf439
# ╠═0dacd922-fdf9-4f80-b9e7-f12a68720d79
# ╟─d17aa551-9b9e-48a2-a699-24c2371fdb85
# ╟─cb6a3f3a-6b9c-4944-8ab0-870ee754ca35
# ╟─58d369b4-c52b-4af9-be3a-955edd253bf6
# ╟─00e1d80f-893e-47dd-8592-faeb71cb4744
# ╟─235031fa-9d28-4398-becc-559eb59c4bd5
# ╟─03bcdf8a-6ef4-4056-a411-937e89c40789
# ╟─5bac1c45-1b54-455a-9150-25e5d65c9008
# ╟─5b205537-307a-4f1e-9739-62b1e1929467
# ╟─d84473d0-f94d-411e-950b-f73ebb5cce37
# ╠═1b2b8427-c417-4d72-998b-ecf6f3687a15
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002