# Unconditionally stable numerical method for 1st order non-linear coupled ODEs?

I am attempting to numerically solve the following system of ODEs:

$$\begin{gather}\frac{dT_1}{dt} = f_1(T_1,T_2), \quad T_1(t=0)=T_{1,0} \\[3pt] \frac{dT_2}{dt} = f_2(T_1,T_2), \quad T_2(t=0)=T_{2,0}\end{gather}$$

The functions $$f_1$$ and $$f_2$$ have $$T$$ and $$T^4$$ terms and therefore are non-linear. I tried the forward Euler method but this becomes unstable even for very small $$\Delta t$$.

Is there any unconditionally stable method of numerical integration to solve this system?

• Is there a connection to physics? – G. Smith Mar 17 '20 at 0:26
• An unconditionally stable method for any type of nonlinear ODE doesn't exist. On the other hand, forward Euler is about the worst possible method for solving any ODE. Either learn something about numerical methods, or use software like MATLAB or Mathematica. – alephzero Mar 17 '20 at 0:38
• @G.Smith Presumably, heat transfer with Newton's law of cooling and radiation? – alephzero Mar 17 '20 at 0:43
• @G.Smith: yes, a solid sphere at temperature $T_2(t)$ is enclosed in a hollow sphere at temperature $T_1(t)$. The hollow sphere at $T_2(t)$ is enclosed in another hollow sphere held at room temperature. The solid sphere and former hollow sphere are being cooling down using two separate refrigerators with known cooling capacity. I want to calculate the cooldown profiles $T_1(t)$ and $T_2(t)$. $T$ and $T^4$ arise from conduction and radiation heat transfer. – cryonole Mar 17 '20 at 0:44
• Have you tried using something like RK4? – Triatticus Mar 17 '20 at 0:51