Numerical Analysis/stability of RK methods/quizzes

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{ the A-stability is characterized by: |type="( )"} - all the points in the left half of the complex plane . - it a special case of the absolute stability. - inludes all $\{z\in ℂ|\mathrm{R}\mathrm{e}(z)<0\}$, + all of the above

{Euler method is: |type="( )"} - A-stable + not A-stable

{the absolute stability region for RK4 is greater than the A-stability region for the same method: |type="()"}

+ false - True - True for certain values of $b_j$.

{A numerical method is stable if : |type="()"} + 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.

{$\lambda =1$ is always a root of the characteristic polynomial of the multistep method: |type="( )"}

+ True - False - usually not the case

{the root of the C.P. can be real or complex,and the method still be stable: |type="( )"} + True - False

{A multi- step method is strongly stable if : |type="( )"} - $\lambda =1$ 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 .

{if more than one root has magnitude equal to 1, and the others are less than one, the method is |type="( )"} - strongly stable. - A-stable + weakly stable. - Unstable.

{The numerical method is unstable if |type="( )"}

- $\lambda >1$ for at least one root - $Re(\lambda )>1$ for at least one root - $Re(\lambda )⩽1$ for at least one root + the firs and the second sentences.

{the absolute stabity region for the explicit Euler's method is |type="( )"} + 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.

{ None of the RK methods is A-Stable: |type="( )"} + True - False

{All explicit methods are : |type="( )"} - A-Stable + Not A-Stable. - Unstable. - None of the above

{implicit multistep methods are A-stable if the have order at most |type="( )"} -1 +2 -3 -they are always A-stable.

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