GATE ME 2021 Set 1 — Question 14
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Engineering Mathematics → Numerical Methods → Euler & Runge-Kutta Methods
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Question
The ordinary differential equation subject to an initial condition is solved numerically using the following scheme:where is the time step, , and . This numerical scheme is stable for all values of in the interval ________.
Correct answer
(A) 0 < h < (2)/(π)
Solution
To find the interval of the time step for which the numerical scheme is stable, we analyze the given finite difference equation:\frac{y(t_{n+1}) - y(t_n)}{h} = -$\pi$ y(t_n)This is the Forward Euler (explicit) discretization of the ordinary differential equation. Let denote the numerical approximation of . Rearranging the equation to express in terms of :y_{n+1} - y_n = -$\pi$ h y_ny_{n+1} = (1 - $\pi$ h) y_nFor the numerical scheme to be stable, the amplification factor must satisfy the stability criterion:|$\lambda$| < 1Substituting the expression for :|1 - $\pi$ h| < 1This inequality can be written as:-1 < 1 - $\pi$ h < 1Subtracting from all parts of the inequality:-2 < -$\pi$ h < 0Dividing by (which reverses the inequality signs):0 < h < \frac{2}{$\pi$}Thus, the numerical scheme is stable for all values of in the interval .Correct Option: A
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