GATE ME 2023 Set 1 — Question 50
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Engineering Mathematics → Numerical Methods → Euler & Runge-Kutta Methods
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Question
The initial value problemis solved numerically using the forward Euler’s method with a constant and positive time step of .
Let represent the numerical solution obtained after steps. The condition is satisfied if and only if does not exceed _____________.
Let represent the numerical solution obtained after steps. The condition is satisfied if and only if does not exceed _____________.
Correct answer
0.999 to 1.001
Solution
The forward Euler method for the ODE is given by:
Here, . Substituting this into the Euler formula:
The condition for stability (non-increasing solution magnitude) is , which implies:
Subtracting 1 from all sides:
Dividing by -2 (and reversing the inequalities):
Since is a positive time step, the condition is . Thus, must not exceed 1.
Here, . Substituting this into the Euler formula:
The condition for stability (non-increasing solution magnitude) is , which implies:
Subtracting 1 from all sides:
Dividing by -2 (and reversing the inequalities):
Since is a positive time step, the condition is . Thus, must not exceed 1.
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