User supplied jacobian makes ode23s very slow

[Alexander Bessman]

Hello,

I'm using ode23s from odepkg to solve a simple 1D PDE, ∂u/∂t = ∂²u/dx².
I have discretized it via the forward-time central-space scheme,
yielding a tridiagonal system of ODEs, which ode23s has no problem
solving. However, when I supply the system's Jacobian via odeset, the
solution takes much longer to complete. The solution is still correct,
but I would have expected it to be faster, not slower. I've included my
code below. The result of running it is thus:

|Timesteps             |CPU-time            |Max stepsize        |
|N  |with Jac |w/o Jac |with Jac  |w/o Jac  |with Jac  |w/o Jac  |
|10 |432      |79      |1.3801 s  |1.1953 s |1.9914e-2 |2.0000e-1|
|20 |1633     |98      |5.4225 s  |2.6037 s |5.2711e-3 |1.9447e-1|
|40 |6442     |119     |26.7256 s |5.9159 s |8.4338e-4 |1.7272e-1|

I'd be grateful of someone could help me understand this phenomenon.
//Alexander Bessman

-----------------------------------------------------------------------

pkg load odepkg

cpu = zeros(2,3);
steps = zeros(2,3);
htmax = zeros(2,3);
iter = 0;
for N = [10, 20, 40]    
    % Jacobian
    e = ones (N, 1);
    jac = spdiags ([e, -2*e, e], [-1:1], N, N);
    jac(end-1, end) = 2;
    conf = odeset ('Jacobian', jac);
    
    u0 = zeros(N,1);
    iter = iter + 1;
    
    tic
    [t, y] = ode23s (@heat_eq, [0, 2], u0);
    cpu(1, iter) = toc;
    steps(1, iter) = length(t);
    htmax(1, iter) = max(diff(t));
    
    tic
    [t, y] = ode23s (@heat_eq, [0, 2], u0, conf);
    cpu(2, iter) = toc;
    steps(2, iter) = length(t);
    htmax(2, iter) = max(diff(t));
end

function dudt = heat_eq (t, u, kappa=1)
    N = length(u);
    e = ones (N, 1);
    A = spdiags ([e, -2*e, e], [-1:1], N, N);
    
    % Boundary conditions
    A(end-1, end) = 2;
    b = zeros(N, 1);
    if t <= 1
        b(1) = 1;
    end
    
    dudt = kappa * N^2 * (A*u + b);