Practice Stability Of Common Methods (18.4) - Stability and Convergence of Methods

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6 sample questions

Question 1.
What is the stability function for Euler's Method?
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Think about the general form of stability functions.
Question 2.
Define A-stability.
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What does it mean for a method to handle certain values of λ?
3Question 3.
What defines the condition of a method being A-stable?
- It is stable for Re(λ) < 0
- It requires large step sizes
- It is stable for Re(λ) > 0
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Think about the real part of λ and its relation to A-stability.
4Question 4.
True or False: All numerical methods are conditionally stable.
- True
- False
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Consider what makes a numerical method reliable.
Question 5.
Assess the stability of the Midpoint method if h = 1 and λ = -3. Show your calculations clearly.
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Recall how to derive R(z) for the midpoint method.
Question 6.
Discuss the impact of the Lax Equivalence Theorem on the design of numerical methods.
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What does this mean for method selection when solving ODEs?

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