Euler’s Method — Question 3

Question 3 – Solution

Strategy. Analyze the discrete amplification factor rather than transferring the differential equation’s stability directly to its numerical update.

Step 1: Solve the recurrence. The update is $Y_{n+1}=(1-4h)Y_n$, so $\boxed{Y_n=(1-4h)^n}$. It tends to zero exactly when $$|1-4h|<1,\qquad \boxed{0<h<1/2}.$$

Step 2: Impose positivity and monotonicity. All values are nonnegative and nonincreasing exactly when $0\le 1-4h<1$, or $$\boxed{0<h\le 1/4}.$$ At $h=1/4$, the approximation becomes zero after one step, although $e^{-4t}$ is strictly positive at every finite time. At $h=1/2$, $Y_n=(-1)^n$ oscillates without decay. For $h>1/2$, its magnitude grows without bound.

Step 3: Compare two oscillating sequences. $$\begin{array}{c|rrrrr} n&0&1&2&3&4\\\hline h=0.4&1&-0.6&0.36&-0.216&0.1296\\ h=0.6&1&-1.4&1.96&-2.744&3.8416 \end{array}$$ Each row is plotted at its own times $t_n=nh$, not at common node indices interpreted as time.

Step 4: Interpret numerical stability. The exact solution $y=e^{-4t}$ decays smoothly. Euler replaces its per-step factor $e^{-4h}$ by $1-4h$, which can be negative or have magnitude exceeding one. At $h=0.4$ the method is asymptotically decaying but produces unphysical sign changes; at $h=0.6$ it also loses decay. Stability controls long-run amplification, while positivity is an additional property with a stricter step requirement.