MATLAB: Solving coupled ODE’s by ode45

ode45 rapid rates stalling

Hi
I have a relatively simple set of coupled ODE's that I am trying to solve by ODE45. The following is the content of the .m-file, which shows the coupled system:
function xprime = eoms(t, x)
xprime = [
    1e9 + 5.0e4*x(3) - 50*x(1);
    4.0e1*x(1) - 3.3e3*x(2);
    2.0e3*x(2) - 5e4*x(3) + 3.5e7*heaviside(t-1)*x(4);
    1.0e3*x(2) - heaviside(t-1)*5.0e7*x(4)];
I solve it using the following command:
x0 = [0 0 0 0];
tspan = [0, 2];
[t, x] = ode45(@eoms, tspan, x0);
However when I compile MatLAB just keeps calculating, it doesn't give me a result. I think it has something to due with the fact that the Heaviside step-function, because if I take it out then it nicely solves the remaining part. I have also tried in a competing software, and the same thing happens (actually, there the solver just gives an error).
Maybe it is also due to the very rapid rates in the equations. Anyhow, I am really lost on what to do – unfortunately I need the transient behavior, so I can't just assume steady-state for some of the equations.
Do I have any options here?
Thanks in advance.
Best, Niles.

Best Answer

You have a ‘stiff’ system, and ode45 is not the best option for it, although it's an appropriate initial experiment. If ode45 seems not to work (and the system otherwise seems to be solvable in some situations as yours was), switching to ode15s is appropriate, especially in systems with coefficients of widely-ranging magniudes such as yours are (from 4E+1 to 1E+9).
After experiencing the same problem you did with ode45, I ran your code with ode15s and it solved your eoms system in 1.2225 seconds.
So I suggest you simply change your ODE function call to:
[t, x] = ode15s(@eoms, tspan, x0);
and enjoy the results!
Also, I coded your function as an implicit function to run it, avoiding the need to save a separate function m-file. (This isn't always possible, but it works in many situations.)
The function:
eoms = @(t,x) [1e9 + 5.0e4*x(3) - 50*x(1);  4.0e1*x(1) - 3.3e3*x(2);  2.0e3*x(2) - 5e4*x(3) + 3.5e7*heaviside(t-1)*x(4);  1.0e3*x(2) - heaviside(t-1)*5.0e7*x(4)];
and the call to it changes to:
[t, x] = ode15s(eoms, tspan, x0);
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