Euler method

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Euler's method numerically approximates solutions of first-order ordinary differential equations (ODEs) with a given initial value. It is an explicit method for solving initial value problems (IVPs), as described in the wikipedia page.

The ODE has to be provided in the following form:

• $\frac{dy(t)}{dt} = f(t,y(t))$

with an initial value

• $y(t_0) = y_0$

To get a numeric solution, we replace the derivative on the LHS with a finite difference approximation:

• $\frac{dy(t)}{dt} \approx \frac{y(t+h)-y(t)}{h}$

then solve for $y(t+h)$:

• $y(t+h) \approx y(t) + h \, \frac{dy(t)}{dt}$

which is the same as

• $y(t+h) \approx y(t) + h \, f(t,y(t))$

The iterative solution rule is then:

• $y_{n+1} = y_n + h \, f(t_n, y_n)$

where $h$ is the step size, the most relevant parameter for accuracy of the solution. A smaller step size increases accuracy but also the computation cost, so it has always has to be hand-picked according to the problem at hand.

Example: Newton's Cooling Law

Newton's cooling law describes how an object of initial temperature $T(t_0) = T_0$ cools down in an environment of temperature $T_R$:

• $\frac{dT(t)}{dt} = -k \, \Delta T$

or

• $\frac{dT(t)}{dt} = -k \, (T(t) - T_R)$

It says that the cooling rate $\frac{dT(t)}{dt}$ of the object is proportional to the current temperature difference $\Delta T = (T(t) - T_R)$ to the surrounding environment.

The analytical solution, which we will compare to the numerical approximation, is

• $T(t) = T_R + (T_0 - T_R) \; e^{-k t}$