18.5 - Example Problems
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Practice Questions
Test your understanding with targeted questions
What does stability in numerical methods refer to?
💡 Hint: Think about what happens to a ship in rough waters.
What is the significance of the Lax Equivalence Theorem?
💡 Hint: Remember the relationship between stability and convergence.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the stability condition for a numerical method?
💡 Hint: Remember, if the stability function is greater than 1, errors will grow.
True or False: A stable method guarantees convergence.
💡 Hint: Consider the Lax Equivalence Theorem.
2 more questions available
Challenge Problems
Push your limits with advanced challenges
Derive the stability condition for a modified Euler method applied to the equation 𝑦' = -𝜆𝑦, where 𝜆 > 0.
💡 Hint: Start by expressing the numerical iteration and finding the corresponding stability function.
Apply the Backward Euler method to the equation 𝑦' = -2𝑦 and discuss its properties, leveraging A-stability.
💡 Hint: Evaluate the backward difference and its relation to the stability function.
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