rcd.m

function rcd
%RCD Stiff ODE from method of lines on reaction-convection-diffusion PDE.

N = 38; a = 1; b = 5e-2;
tspan = [0 2]; space = [1:N]/(N+1);

y0 = 0.5*(1+cos(2*pi*space));
y0 = y0(:);
options = odeset('Jacobian',@jacobian);
options = odeset(options,'RelTol',1e-3,'AbsTol',1e-3);

[t,y] = ode15s(@f,tspan,y0,options);
e = ones(size(t)); U = [e y e];
waterfall([0:1/(N+1):1],t,U)
colormap hsv
xlabel('space','FontSize',12), ylabel('time','FontSize',12)

    % ------------------------ Nested functions -------------------------
    function dydt = f(t,y)
    %F    Differential equation.

    r1 = -a*(N+1)/2;
    r2 = b*(N+1)^2;
    up = [y(2:N);0]; down = [0;y(1:N-1)];
    e1 = [1;zeros(N-1,1)]; eN = [zeros(N-1,1);1];

    dydt = r1*(up-down) + r2*(-2*y+up+down) + (r2-r1)*e1 + ...
           (r2+r1)*eN + y.*(1-y);
    end

    function dfdy = jacobian(t,y)
    %JACOBIAN    Jacobian matrix.

    r1 = -a*(N+1)/2;
    r2 = b*(N+1)^2;
    u = (r2-r1)*ones(N,1);
    v = (-2*r2+1)*ones(N,1) - 2*y;
    w = (r2+r1)*ones(N,1);

    dfdy = spdiags([u v w],[-1 0 1],N,N);
    end

end