Numerical Stability of Euler's Method

Calculus Level 5

Suppose we want to use Euler's Method to numerically (discretely) compute the time-equation below through a sequence of iterative steps:

\[\large{y = e^{-A t} \\ y_k = y_{k-1} + y'_{k-1} \Delta t}\]

In the above equation, \(\Delta t = \frac{1}{1000}\). \(y_k\) denotes the present value of the function, \(y_{k-1}\) denotes the previous value of the function, and \(y'_{k-1}\) denotes the previous value of the function's time-derivative. The parameter \(A\) is an arbitrary positive constant.

Suppose \(y_0 = 1\) and \(y'_0 = -A\).

For \(0 < A < A_1\), the system undergoes monotonic decay toward zero.
For \(A_1 < A < A_2\), the system undergoes damped oscillation.
For \(A > A_2\), the system undergoes divergent oscillation.

Determine \(A_1 + A_2\). Assume no rounding or truncation errors (we have a perfect computer).

Note: This version of Euler's Method is more specifically known as the "forward" or "explicit" version.

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