Differential Equations Numerical Methods for Differential Equations Flashcards

24 sample cards from the Numerical Methods for Differential Equations deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. How is stiffness characterized for a linear system $\vec{y}' = A\vec{y}$?

   By a large stiffness ratio: $\dfrac{\max_i |\operatorname{Re}(\lambda_i)|}{\min_i |\operatorname{Re}(\lambda_i)|} \gg 1$, where $\lambda_i$ are eigenvalues of $A$ (all with negative real part).

2. Define the region of absolute stability of a numerical method.

   The set of complex values $z = h\lambda$ for which the method applied to the test equation $y' = \lambda y$ produces a bounded (non-growing) numerical solution.

3. What is the test equation used to analyze numerical stability?

   $$y' = \lambda y, \qquad \lambda \in \mathbb{C}$$ whose exact solution $y=y_{0}e^{\lambda x}$ decays when $\operatorname{Re}(\lambda)<0$.

4. What is the stability condition for forward Euler applied to $y'=\lambda y$ with real $\lambda<0$?

   The amplification factor is $1+h\lambda$, so stability requires $|1+h\lambda|\le 1$, i.e. $h \le \dfrac{2}{|\lambda|}$ (conditionally stable).

5. Define A-stability.

   A method is A-stable if its region of absolute stability contains the entire left half of the complex plane ($\operatorname{Re}(z)<0$), so it is stable for any $h>0$ on all decaying test problems — ideal for stiff equations.

6. Are explicit Runge-Kutta methods ever A-stable? What is preferred for stiff systems?

   No explicit RK (or any explicit) method can be A-stable. Implicit methods — backward Euler, implicit RK, BDF (Gear) methods — are used for stiff systems.

7. Define the consistency of a numerical method for ODEs.

   A method is consistent if its local truncation error tends to zero faster than $h$ as $h\to 0$, i.e. the difference formula approaches the differential equation; equivalently the LTE is $o(h)$ (order $\ge 1$).

8. State Dahlquist's equivalence theorem (Lax-type) relating consistency, stability, and convergence.

   For a linear multistep method, consistency + zero-stability $\iff$ convergence. That is, a consistent method converges if and only if it is zero-stable.

9. What is zero-stability (root condition) for a linear multistep method?

   The roots of the method's first characteristic polynomial $\rho(\zeta)$ must lie inside or on the unit circle, with any root on the unit circle being simple. This bounds error growth as $h\to 0$.

10. Distinguish local truncation error from global error.

    Local truncation error is the error introduced in a single step assuming exact prior values. Global error is the total accumulated error at a point after many steps; for a method of local order $p+1$ the global order is $p$.

11. State the first Dahlquist barrier for linear multistep methods.

    A zero-stable $k$-step linear multistep method can have order at most $k+1$ for odd $k$ and $k+2$ for even $k$ ($k+2$ only if it is not... the maximum attainable order of a stable $k$-step method is $k+2$).

12. How is a higher-order ODE converted for numerical solution? Give the reduction for $y'' = g(x, y, y')$.

    Introduce $u_{1}=y$, $u_{2}=y'$ to get the first-order system $$u_{1}' = u_{2}, \qquad u_{2}' = g(x, u_{1}, u_{2})$$ then apply a standard method to the vector system.

13. How is a numerical method like RK4 applied to a system of first-order ODEs $\vec{y}' = \vec{f}(x,\vec{y})$?

    Identically to the scalar case but with vector quantities: each $k_{i}$ becomes a vector $\vec{k}_{i}=\vec{f}(x_{n}+c_i h,\, \vec{y}_n + \dots)$, and the update advances all components simultaneously with the same $h$.

14. For an $n$-th order ODE, how many first-order equations and initial conditions are needed?

    It converts to a system of $n$ first-order ODEs requiring $n$ initial conditions: $y(x_{0}), y'(x_{0}), \dots, y^{(n-1)}(x_{0})$.

15. What type of problem does the Shooting Method solve, and what is its basic idea?

    It solves boundary value problems (BVPs). Idea: convert the BVP to an IVP by guessing the unknown initial slope, integrate, and iteratively adjust the guess until the far boundary condition is satisfied.

16. In the shooting method for $y''=f(x,y,y')$ with $y(a)=\alpha,\ y(b)=\beta$, what root-finding problem is solved?

    Let $y(b;s)$ be the endpoint value from initial slope $y'(a)=s$. Solve $\phi(s) = y(b;s) - \beta = 0$ for $s$, typically by the secant method or Newton's method.

17. For a linear BVP, how does the shooting method obtain the exact solution from two guesses?

    Because the endpoint depends linearly on the initial slope, shoot with two guesses $s_{0}, s_{1}$ and linearly interpolate: the correct slope is $$s = s_{0} + (s_{1}-s_{0})\frac{\beta - y(b;s_{0})}{y(b;s_{1}) - y(b;s_{0})}$$

18. What is the basic idea of the Finite Difference Method (FDM) for BVPs?

    Discretize the domain into a grid and replace derivatives with finite-difference approximations, converting the differential equation into a system of algebraic equations at the grid points.

19. Give the central difference approximations for $y'(x_i)$ and $y''(x_i)$ on a uniform grid of spacing $h$.

    $$y'(x_i) \approx \frac{y_{i+1}-y_{i-1}}{2h}, \qquad y''(x_i) \approx \frac{y_{i+1}-2y_{i}+y_{i-1}}{h^{2}}$$ both with error $O(h^{2})$.

20. Applying central differences to $y'' = f(x)$ with Dirichlet conditions yields what kind of linear system?

    A tridiagonal linear system $A\vec{y}=\vec{b}$, which can be solved efficiently with the Thomas algorithm.

21. Give the forward, backward, and central first-difference formulas and their orders of accuracy.

    Forward: $\frac{y_{i+1}-y_i}{h}$, $O(h)$. Backward: $\frac{y_i-y_{i-1}}{h}$, $O(h)$. Central: $\frac{y_{i+1}-y_{i-1}}{2h}$, $O(h^{2})$.

22. Compare the shooting method and the finite difference method for BVPs.

    Shooting reduces the BVP to IVPs solved with iteration on the initial slope (simple, but can be unstable for sensitive problems). FDM solves all grid unknowns simultaneously as one linear/nonlinear system (more stable for stiff/sensitive BVPs, but larger to set up).

23. What is the order of convergence of a numerical method, and how is it observed experimentally?

    A method has order $p$ if the global error behaves like $E(h) \approx C h^{p}$. Experimentally, halving $h$ reduces the error by a factor of about $2^{p}$; plotting $\log E$ vs $\log h$ gives slope $p$.

24. Why can round-off error dominate truncation error when $h$ is made too small?

    Truncation error decreases with $h$, but round-off error (from finite-precision arithmetic, e.g. dividing by tiny $h$) grows as $h\to 0$, so total error has a minimum at an optimal $h$; going below it degrades accuracy.

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