Numerical Differential Equations

From MGSA

State precisely the root condition for $\sum(a_jy_{n+j})=h\sum{f(x_{n+j},y_{n+j})}$ .
Consider ${d\over dt}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}-10^6&0\\ 0&1\end{pmatrix} \begin{pmatrix}x \\ y \end{pmatrix}$ . Would Runge-Kutta be reasonable to use? Do you anticipate any problems?
Define stability for the general scheme $u n + 1 = Q u n$ . Define consistency and convergence.
Consider the PDE $v t = v x$ , $v (x,0)$ given, and the numerical scheme ${u_j^{n+1}-u_j^n\over k}={u_{j+1}^n - u_j\over h}$ . Is this scheme consistent? Stabe? What is the CFL condition and what does it say about $k$ and $h$ ?
Prove the Fourier transform of $D + u n$ is ${1\over h}(e^{-i\xi}-1)\hat{u^n}$ .
Given a scheme $u_j^{n+1}=\alpha u_j^n-2u_{j+1}^n + u_{j+1}^n$ , describe how you would find values of $α$ for which it is stable.
Assume $\hat{u^{n+1}}=\rho(\xi)\hat{u^n}$ and $|\rho(\xi)|\le 1$ . Show that his implies stability.
What is the advantage/disadvantage of RK methods comparing to multi-step methods?
State the root condition for RK methods.
Draw the stability regions for the 2 and 3 step BDF methods.
State the difference between the Barrier and Dahlquist Barrier theorems. Whats the highest order attainable for an s-step multistep method?

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