# 2022-09-28 Runge-Kutta#

## Last time#

• $$A$$- and $$L$$-stability

• Spatial, temporal, and physical dissipation

• Stiffness

## Today#

• Exploring stiffness

• Runge-Kutta methods

using Plots
default(linewidth=3)
using LinearAlgebra
using SparseArrays

function plot_stability(Rz, title; xlims=(-3, 3), ylims=(-2, 2))
x = LinRange(xlims[1], xlims[2], 100)
y = LinRange(ylims[1], ylims[2], 100)
heatmap(x, y, (x, y) -> abs(Rz(x + 1im*y)), c=:bwr, clims=(0, 2), aspect_ratio=:equal, title=title)
end

function ode_theta_linear(A, u0; forcing=zero, tfinal=1, h=0.1, theta=.5)
u = copy(u0)
t = 0.
thist = [t]
uhist = [u0]
while t < tfinal
tnext = min(t+h, tfinal)
h = tnext - t
rhs = (I + h*(1-theta)*A) * u .+ h*forcing(t+h*theta)
u = (I - h*theta*A) \ rhs
t = tnext
push!(thist, t)
push!(uhist, u)
end
thist, hcat(uhist...)
end

Rz_theta(z, theta) = (1 + (1 - theta)*z) / (1 - theta*z)

dx = 2 / n
rows = [1]
cols = [1]
vals = [0.]
wrap(j) = (j + n - 1) % n + 1
for i in 1:n
append!(rows, [i, i, i])
append!(cols, wrap.(i-1:i+1))
diffuse = [-1, 2, -1] * kappa / dx^2
advect_upwind = [-1, 1, 0] * wind / dx
advect_center = [-1, 0, 1] * wind / 2dx
stencil = -diffuse - upwind * advect_upwind - (1 - upwind) * advect_center
append!(vals, stencil)
end
sparse(rows, cols, vals)
end

5×5 SparseMatrixCSC{Float64, Int64} with 15 stored entries:
-1.25   -0.625    ⋅       ⋅      1.875
1.875  -1.25   -0.625    ⋅       ⋅
⋅      1.875  -1.25   -0.625    ⋅
⋅       ⋅      1.875  -1.25   -0.625
-0.625    ⋅       ⋅      1.875  -1.25


# Stiffness#

Stiff equations are problems for which explicit methods don’t work. (Hairer and Wanner, 2002)

• “stiff” dates to Curtiss and Hirschfelder (1952)

k = 5
thist, uhist = ode_theta_linear(-k, [.2], forcing=t -> k*cos(t), tfinal=5, h=.5, theta=1)
scatter(thist, uhist[1,:])
plot!(cos)

function mms_error(h; theta=.5, k=10)
u0 = [.2]
thist, uhist = ode_theta_linear(-k, u0, forcing=t -> k*cos(t), tfinal=3, h=h, theta=theta)
T = thist[end]
u_exact = (u0 .- k^2/(k^2+1)) * exp(-k*T) .+ k*(sin(T) + k*cos(T))/(k^2 + 1)
uhist[1,end] .- u_exact
end

mms_error (generic function with 1 method)

hs = .5 .^ (1:8)
errors = mms_error.(hs, theta=1, k=1000)
plot(hs, norm.(errors), marker=:circle, xscale=:log10, yscale=:log10)
plot!(hs, hs, label="\$h\$", legend=:topleft)
plot!(hs, hs.^2, label="\$h^2\$")


theta=0
n = 80
dx = 2 / n
kappa = .5
lambda_min = -4 * kappa / dx^2
cfl = 1
h = min(-2 / lambda_min, cfl * dx)

plot_stability(z -> Rz_theta(z, theta),


# Heun’s method and RK4#

heun = RKTable([0 0; 1 0], [.5 .5])
plot_stability(z -> rk_stability(z, heun), "Heun's method")

rk4 = RKTable([0 0 0 0; .5 0 0 0; 0 .5 0 0; 0 0 1 0],
[1 2 2 1] / 6)
plot_stability(z -> rk_stability(z, rk4), "RK4")


# An explicit RK solver#

(19)#\begin{align} Y_i &= u(t) + h \sum_j a_{ij} f(t+c_j h, Y_j) & u(t+h) &= u(t) + h \sum_j b_j f(t+c_j h, Y_j) \end{align}
function ode_rk_explicit(f, u0; tfinal=1, h=0.1, table=rk4)
u = copy(u0)
t = 0.
n, s = length(u), length(table.c)
fY = zeros(n, s)
thist = [t]
uhist = [u0]
while t < tfinal
tnext = min(t+h, tfinal)
h = tnext - t
for i in 1:s
ti = t + h * table.c[i]
Yi = u + h * sum(fY[:,1:i-1] * table.A[i,1:i-1], dims=2)
fY[:,i] = f(ti, Yi)
end
u += h * fY * table.b[1,:]
t = tnext
push!(thist, t)
push!(uhist, u)
end
thist, hcat(uhist...)
end

ode_rk_explicit (generic function with 1 method)

linear_oscillator(t, u) = [0 -1; 1 0] * u
thist, uhist = ode_rk_explicit(linear_oscillator, [1., 0], tfinal=6, h=1.2)
plot(thist, uhist', marker=:auto)
plot!([sin, cos])


# Measuring convergence and accuracy#

function mms_error(h, f, u_exact; table=rk4)
u0 = u_exact(0)
thist, uhist = ode_rk_explicit(f, u0, tfinal=3, h=h, table=table)
T = thist[end]
uhist[:,end] - u_exact(T)
end

hs = .5 .^ (-3:8)
linear_oscillator_exact(t) = [cos(t), sin(t)]

linear_oscillator_exact (generic function with 1 method)

heun_errors = mms_error.(hs, linear_oscillator, linear_oscillator_exact, table=heun)
rk4_errors = mms_error.(hs, linear_oscillator, linear_oscillator_exact, table=rk4)
plot(hs, [norm.(heun_errors) norm.(rk4_errors)], label=["heun" "rk4"], marker=:auto)
plot!(hs, [hs hs.^2 hs.^3 hs.^4], label=["\$h^$p\\$" for p in [1 2 3 4]], legend=:topleft, xscale=:log10, yscale=:log10)


# Work-precision (or accuracy vs cost)#

heun_nfeval = length(heun.c) ./ hs
rk4_nfeval = length(rk4.c) ./ hs
plot(heun_nfeval, norm.(heun_errors), marker=:auto, label="heun")
plot!(rk4_nfeval, norm.(rk4_errors), marker=:auto, label="rk4")
plot!(xscale=:log10, yscale=:log10, xlabel="number of evaluations", ylabel="error")


# Effective RK stability diagrams#

function rk_eff_stability(z, rk; brow=1)
s = size(rk.b, 2)
fsal = rk.b[1,:] ≈ rk.A[end,:]
z = (s - fsal) * z
1 + z * rk.b[brow,:]' * ((I - z*rk.A) \ ones(s))
end
plot_stability(z -> rk_eff_stability(z, heun), "Heun's method")

plot_stability(z -> rk_eff_stability(z, rk4), "RK4", xlims=(-2, 2), ylims=(-2, 2))


# Runge-Kutta order conditions#

We consider the autonomous differential equation

$\dot u = f(u) .$

Higher derivatives of the exact soultion can be computed using the chain rule, e.g.,

\begin{align*} \ddot u(t) &= f'(u) \dot u = f'(u) f(u) \\ \dddot u(t) &= f''(u) f(u) f(u) + f'(u) f'(u) f(u) . \\ \end{align*}

Note that if $$f(u)$$ is linear, $$f''(u) = 0$$. Meanwhile, the numerical solution is a function of the time step $$h$$,

$\begin{split}\begin{split} Y_i(h) &= u(0) + h \sum_j a_{ij} f(Y_j) \\ U(h) &= u(0) + h \sum_j b_j f(Y_j). \end{split}\end{split}$

We will take the limit $$h\to 0$$ and equate derivatives of the numerical solution. First we differentiate the stage equations,

\begin{split} Y_i(0) &= u(0) \ \dot Y_i(0) &= \sum_j a_{ij} f(Y_j) \ \ddot Y_i(0) &= 2 \sum_j a_{ij} \dot f(Y_j) \ &= 2 \sum_j a_{ij} f’(Y_j) \dot Y_j \ &= 2\sum_{j,k} a_{ij} a_{jk} f’(Y_j) f(Y_k) \ \dddot Y_i(0) &= 3 \sum_j a_{ij} \ddot f (Y_j) \ &= 3 \sum_j a_{ij} \Big( \sum_k f’’(Y_j) \dot Y_j \dot Y_k + f’(Y_j) \ddot Y_j \Big) \ &= 3 \sum_{j,k,\ell} a_{ij} a_{jk} \Big( a_{j\ell} f’’(Y_j) f(Y_k) f(Y_\ell) + 2 a_{k\ell} f’(Y_j) f’(Y_k) f(Y_\ell) \Big) \end{split}

where we have used Liebnitz’s formula for the $$m$$th derivative,

$(h \phi(h))^{(m)}|_{h=0} = m \phi^{(m-1)}(0) .$
Similar formulas apply for $$\dot U(0)$$, $$\ddot U(0)$$, and $$\dddot U(0)$$, with $$b_j$$ in place of $$a_{ij}$$.

# Order conditions are nonlinear algebraic equations#

Equating terms $$\dot u(0) = \dot U(0)$$ yields

$\sum_j b_j = 1,$
equating $$\ddot u(0) = \ddot U(0)$$ yields
$2 \sum_{j,k} b_j a_{jk} = 1 ,$
and equating $$\dddot u(0) = \dddot U(0)$$ yields the two equations \begin{split} 3\sum_{j,k,\ell} b_j a_{jk} a_{j\ell} &= 1 \ 6 \sum_{j,k,\ell} b_j a_{jk} a_{k\ell} &= 1 . \end{split}

# Observations#

• These are systems of nonlinear equations for the coefficients $$a_{ij}$$ and $$b_j$$. There is no guarantee that they have solutions.

• The number of equations grows rapidly as the order increases.

$$u^{(1)}$$

$$u^{(2)}$$

$$u^{(3)}$$

$$u^{(4)}$$

$$u^{(5)}$$

$$u^{(6)}$$

$$u^{(7)}$$

$$u^{(8)}$$

$$u^{(9)}$$

$$u^{(10)}$$

# terms

1

1

2

4

9

20

48

115

286

719

cumulative

1

2

4

8

17

37

85

200

486

1205

• Usually the number of order conditions does not exactly match the number of free parameters, meaning that the remaining parameters can be optimized (usually numerically) for different purposes, such as to minimize the leading error terms or to maximize stability in certain regions of the complex plane. Finding globally optimal solutions can be extremely demanding.

• Rooted trees provide a convenient notation

# Theorem (from Hairer, Nørsett, and Wanner)#

A Runge-Kutta method is of order $$p$$ if and only if

$\gamma(\mathcal t) \sum_{j} b_j \Phi_j(t) = 1$
for all trees $$t$$ of order $$\le p$$.

For a linear autonomous equation

$\dot u = A u$
we only need one additional order condition per order of accuracy because $$f'' = 0$$. These conditions can also be derived by equating derivatives of the stability function $$R(z)$$ with the exponential $$e^z$$. For a linear non-autonomous equation
$\dot u = A(t) u + g(t)$
or more generally, an autonomous system with quadratic right hand side,
$\dot u = B (u \otimes u) + A u + C$
where $$B$$ is a rank 3 tensor, we have $$f''' = 0$$, thus limiting the number of order conditions.

# Embedded error estimation#

It is often possible to design Runge-Kutta methods with multiple completion orders, say of order $$p$$ and $$p-1$$.

$\begin{split}\left[ \begin{array}{c|c} c & A \\ \hline & b^T \\ & \tilde b^T \end{array} \right] . \end{split}$

The classical RK4 does not come with an embedded method, but most subsequent RK methods do. The Bogacki-Shampine method is one such method.

bs3 = RKTable([0 0 0 0; 1/2 0 0 0; 0 3/4 0 0; 2/9 1/3 4/9 0],
[2/9 1/3 4/9 0; 7/24 1/4 1/3 1/8])
plot_stability(z -> rk_eff_stability(z, bs3), "BS3 third order")

plot_stability(z -> rk_eff_stability(z, bs3, brow=2), "BS3 second order")


# Properties#

display(bs3.A)
display(bs3.b)

4×4 Matrix{Float64}:
0.0       0.0       0.0       0.0
0.5       0.0       0.0       0.0
0.0       0.75      0.0       0.0
0.222222  0.333333  0.444444  0.0

2×4 Matrix{Float64}:
0.222222  0.333333  0.444444  0.0
0.291667  0.25      0.333333  0.125

• First same as last (FSAL): the completion formula $$b$$ is the same as the last row of $$A$$.

• Stage can be reused as first stage of the next time step. So this 4-stage method has a cost equal to a 3-stage method.

• Fehlberg, a 6-stage, 5th order method for which the 4th order embedded formula has been optimized for accuracy.

• Dormand-Prince, a 7-stage, 5th order method with the FSAL property, with the 5th order completion formula optimized for accuracy.

Given a completion formula $$b^T$$ of order $$p$$ and $$\tilde b^T$$ of order $$p-1$$, an estimate of the local truncation error (on this step) is given by

$e_{\text{loc}}(h) = \lVert h (b - \tilde b)^T f(Y) \rVert \in O(h^p) .$
Given a tolerance $$\epsilon$$, we would like to find $$h_*$$ such that
$e_{\text{loc}}(h_*) < \epsilon .$
If
$e_{\text{loc}}(h) = c h^p$
for some constant $$c$$, then
$c h_*^p < \epsilon$
implies
$h_* < \left( \frac{\epsilon}{c} \right)^{1/p} .$
Given the estimate with the current $$h$$,
$c = e_{\text{loc}}(h) / h^p$
we conclude
$\frac{h_*}{h} < \left( \frac{\epsilon}{e_{\text{loc}}(h)} \right)^{1/p} .$

## Notes#

• Usually a “safety factor” less than 1 is included so the predicted error is less than the threshold to reject a time step.

• We have used an absolute tolerance above. If the values of solution variables vary greatly in time, a relative tolerance $$e_{\text{loc}}(h) / \lVert u(t) \rVert$$ or a combination thereof is desirable.

• There is a debate about whether one should optimize the rate at which error is accumulated with respect to work (estimate above) or with respect to simulated time (as above, but with error behaving as $$O(h^{p-1})$$). For problems with a range of time scales at different periods, this is usually done with respect to work.

• Global error control is an active research area.

# Diagonally Implicit Runge-Kutta methods#

$\begin{split} \left[ \begin{array}{c|c} c & A \\ \hline & b^T \end{array} \right] = \left[ \begin{array}{c|cc} c_0 & a_{00} & 0 \\ c_1 & a_{10} & a_{11} \\ \hline & b_0 & b_1 \end{array} \right] \end{split}$
# From Hairer and Wanner Table 6.5 L-stable SDIRK
dirk4 = RKTable([1/4 0 0 0 0; 1/2 1/4 0 0 0; 17/50 -1/25 1/4 0 0; 371/1360 -137/2720 15/544 1/4 0; 25/24 -49/48 125/16 -85/12 1/4],
[24/24 -49/48 125/16 -85/12 1/4; 59/48 -17/96 225/32 -85/12 0]);
#dirk4.A

plot_stability(z -> rk_eff_stability(z, dirk4, brow=1), "DIRK fourth order")

plot_stability(z -> rk_eff_stability(z, dirk4, brow=2), "DIRK third order")


# Fully implicit Runge-Kutta methods#

$\begin{split} \left[ \begin{array}{c|c} c & A \\ \hline & b^T \end{array} \right] = \left[ \begin{array}{c|cc} c_0 & a_{00} & a_{01} \\ c_1 & a_{10} & a_{11} \\ \hline & b_0 & b_1 \end{array} \right] \end{split}$
radau3 = RKTable([5/12 -1/12; 3/4 1/4], [3/4 1/4])

gauss4 = RKTable([1/4 1/4-sqrt(3)/6; 1/4+sqrt(3)/6 1/4],
[1/2 1/2])
plot_stability(z -> rk_eff_stability(z, gauss4), "Gauss 4")


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