## Code for Figure 8.4(b)

### main.m

```
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35% Example numerical integration of a stiff ODE.

% Butcher tableaus (only a and b) for the three methods.
tabs = struct();
tabs.ieu = struct('a', 1, 'b', 1);
tabs.mid = struct('a', 0.5, 'b', 1);
tabs.gl4 = struct('a', [1/4, 1/4 - sqrt(3)/6; 1/4 + sqrt(3)/6, 1/4], ...
'b', [1/2, 1/2]);
tabnames = fieldnames(tabs);

% Run each method.
f = @stiffode;
x0 = [1; 0];
Neval = 10;
T = 2*pi();
x = struct();
for n = 1:length(tabnames)
n = tabnames{n};
Nstep = Neval/length(tabs.(n).b); % Use Neval function evaluations.
h = T/Nstep;
x.(n) = zeros(2, Nstep + 1);
x.(n)(:,1) = x0;
for i = 1:Nstep
x.(n)(:,i + 1) = butcherintegration(x.(n)(:,i), f, h, ...
tabs.(n).a, tabs.(n).b);
end
end

theta = linspace(0, 2*pi(), 257);
x.exact = [cos(theta); -sin(theta)];
tabnames = [{'exact'}; tabnames];

% Make a plot.
errorplot2d(x, T, 'SouthEast');

```

### stiffode.m

```
1
2
3
4
5
6
7
8
9
10function [xdot, jac] = stiffode(x)
% [xdot, jac] = stiffode(x)
%
% Example stiff ODE. First output is dx/dt; second is Jacobian of dx/dt.
A = [0 1; -1 0];
lambda = 500;
rho = (x'*x - 1);
xdot = A*x - lambda*rho*x;
jac = A - eye(2)*lambda*rho - 2*lambda*(x*x');
end%function

```

### butcherintegration.m

```
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93function [xplus, converged, iter, k] = butcherintegration(x, func, h, a, b, Niter)
% [xplus, converged, iter, k] = butcherintegration(x, f, h, a, b, [Niter=100])
%
% Performs numerical integration of f starting from x for h time units.
%
% f must be a function handle that returns dx/dt. If a defines an implicit
% method, then f must also return the Jacobian of dx/dt. In either case, f must
% be time-invariant.
%
% a and b should be the coefficients from the Butcher tableau for the
% integration scheme. If the a matrix defines an explicit scheme (i.e., zero on
% and above the diagonal, then the explicit formulas will be used. Otherwise,
% the implicit equations are solved using Newton's method.
%
% Niter is the maximum number of iterations to perform. xguess is a guess for
% x at the next timestep.
narginchk(5, 6);
if nargin() < 6
Niter = 100;
end

% Check arguments.
x = x(:);
Nx = length(x);
b = h*b(:);
Ns = length(b);
if ~isequal(size(a), [Ns, Ns])
error('a must be a square matrix with a column for each element of b!');
end
a = h*a;

% Decide if tableau is explicit or not.
if nnz(triu(a)) == 0
k = butcher_explicit(x, func, a);
converged = true();
iter = 0;
else
[k, converged, iter] = butcher_implicit(x, func, a, Niter);
end

xplus = x + k*b;

end%function

% *****************************************************************************

function k = butcher_explicit(x, func, a)
% Performs an explicit integration step.
Nx = length(x);
Ns = size(a, 1);
k = zeros(Nx, Ns);
for i = 1:Ns
k(:,i) = func(x + k*a(i,:)');
end
end%function

% *****************************************************************************

function [k, converged, iter] = butcher_implicit(x, func, a, Niter)
% Solves implicit integration using Newton's Method. Timestep is assumed 1.
narginchk(4, 4);
Nx = length(x);
Ns = size(a, 1);;
k = zeros(Nx, Ns);
converged = false();
iter = 0;
while ~converged && iter < Niter
iter = iter + 1;

dX = k*a';
if Ns == 1
[f, J] = func(x + dX);
J = a*J;
else
f = zeros(Nx*Ns, 1);
J = zeros(Nx*Ns, Nx*Ns);
for i = 1:Ns
I = ((i - 1)*Nx + 1):(i*Nx);
[f(I), Ji] = func(x + dX(:,i));
J(I,:) = kron(a(i,:), Ji);
end
end

f = f - k(:);
J = J - eye(Nx*Ns);
dk = -J\f;
k = k + reshape(dk, [Nx, Ns]);

converged = norm(dk) < 1e-8;
end

end%function

```

### errorplot2d.m

```
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61function errorplot2d(x, T, location)
% errorplot2d(x, T, [location])
%
% Makes a plot of integration error over the given time interval T.
%
% x must be a struct with fields to plot. All step sizes are assumed to be
% uniform. If x contains an 'exact' field, it is handled specially.
narginchk(2, 3);
if nargin() < 3
location = {};
else
location = {'location', location};
end

% Create figure and axes.
figure();

ax1 = subplot(2, 2, 1);
hold(ax1, 'on');
ylabel(ax1, 'x_1', 'rotation', 0);
xlabel(ax1, 't');

ax2 = subplot(2, 2, 3);
hold(ax2, 'on');
ylabel(ax2, 'x_2', 'rotation', 0);
xlabel(ax2, 't');

axp = subplot(2, 2, [2, 4]);
hold(axp, 'on');
ylabel(axp, 'x_2', 'rotation', 0);
xlabel(axp, 'x_1');

% Plot exact solution.
if isfield(x, 'exact')
t = linspace(0, T, size(x.exact, 2));
plot(ax1, t, x.exact(1,:), '-k');
plot(ax2, t, x.exact(2,:), '-k');
plot(axp, x.exact(1,:), x.exact(2,:), '-k');
exact = {'exact'};
else
exact = {};
end

% Plot other solutions.
fields = setdiff(fieldnames(x), {'exact'});
markers = {'o', 's', 'd', '^', 'v', '>', '<', 'p', 'h'};
markers = markers(mod(0:(length(fields) - 1), length(markers)) + 1);
colors = hsv(length(fields));
for i = 1:length(fields)
f = fields{i};
t = linspace(0, T, size(x.(f), 2));
style = {markers{i}, 'color', colors(i,:)};
plot(ax1, t, x.(f)(1,:), style{:});
plot(ax2, t, x.(f)(2,:), style{:});
plot(axp, x.(f)(1,:), x.(f)(2,:), style{:});
end