Numerical Analysis/stability of RK methods/quizzes

{ the A-stability is characterized by: - all the points in the left half of the complex plane. - it a special case of the absolute stability. - inludes all $$\{ z \in \mathbb{C} | \mathrm{Re}(z) < 0 \}$$, + all of the above
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{Euler method is: - A-stable + not A-stable
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{the absolute stability region for RK4 is greater than the A-stability region for the same method:
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+ false - True - True for certain values of $$ b_j$$.

{A numerical method is stable if : + small change in the initial condition will produce small change in subsequent steps. - small change in the initial condition will produce huge change in subsequent steps. - big change in the initial conditions produce oscillatory solution at the end.
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{$$ \lambda =1$$ is always a root of the characteristic polynomial of the multistep method:
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+ True - False - usually not the case

{the root of the C.P. can be real or complex,and the method still be stable: + True - False
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{A multi- step method is strongly stable if : - $$ \lambda =1$$ is the only root of magnitude 1. - all other roots has magnitude <1 + the first and the second sentence - just one root inside the unit circle if it has more than one.
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{if more than one root has magnitude equal to 1, and the others are less than one, the method is - strongly stable. - A-stable + weakly stable. - Unstable.
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{The numerical method is unstable if - $$ \lambda > 1$$ for at least one root - $$ Re(\lambda) > 1$$ for at least one root - $$ Re(\lambda) \leqslant 1$$ for at least one root + the firs and the second sentences.
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{the absolute stabity region for the explicit Euler's method is + unit cicle in the complex plane, its center shifted to the left, by one unit. - unit cicle in the complex plane, its center shifted to the rigt, by one unit. - the whole left side of the complex plane. - the method is unstable.
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{ None of the RK methods is A-Stable: + True - False
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{All explicit methods are : - A-Stable + Not A-Stable. - Unstable. - None of the above
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{implicit multistep methods are A-stable if the have order at most -1 +2 -3 -they are always A-stable.
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