Differential Equations Numerical Methods for Differential Equations Syllabus

State the fourth-order Adams-Bashforth method (AB4).

$$y_{n+1} = y_{n} + \frac{h}{24}\left(55 f_{n} - 59 f_{n-1} + 37 f_{n-2} - 9 f_{n-3}\right)$$

Are Adams-Bashforth methods explicit or implicit, and on what are they based?

They are explicit multistep methods based on integrating an interpolating polynomial through past values $f_{n}, f_{n-1}, \dots$ (using only known points).

State the second-order Adams-Moulton method (the trapezoidal rule, AM2).

$$y_{n+1} = y_{n} + \frac{h}{2}\left(f_{n+1} + f_{n}\right)$$ It is implicit because it uses $f_{n+1}=f(x_{n+1},y_{n+1})$.

State the third-order Adams-Moulton method (AM3).

$$y_{n+1} = y_{n} + \frac{h}{12}\left(5 f_{n+1} + 8 f_{n} - f_{n-1}\right)$$

Are Adams-Moulton methods explicit or implicit, and how do they compare to Adams-Bashforth of the same step count?

They are implicit (include $f_{n+1}$). For the same number of steps, Adams-Moulton has smaller error constants and better stability than the explicit Adams-Bashforth method.

How does a predictor-corrector scheme combine Adams-Bashforth and Adams-Moulton methods?

The explicit Adams-Bashforth formula predicts $y_{n+1}^{P}$; this value is used to evaluate $f_{n+1}$, then the implicit Adams-Moulton formula corrects it to give the final $y_{n+1}$, avoiding solving an implicit equation.

What do the acronyms PEC and PECE denote in predictor-corrector schemes?

P = Predict, E = Evaluate $f$, C = Correct. PEC does one correction; PECE additionally re-evaluates $f$ at the corrected value before the next step, improving stability.

How can a predictor-corrector pair provide an error estimate for step-size control?

The difference between the predicted and corrected values $|y_{n+1}^{C} - y_{n+1}^{P}|$ estimates the local truncation error (Milne's device), which can be used to adjust $h$.

What is one key advantage and one disadvantage of multistep methods versus Runge-Kutta methods?

Advantage: fewer function evaluations per step (reuse past $f$ values), so more efficient. Disadvantage: not self-starting (need other methods for initial points) and harder to change step size.

Why are multistep methods described as 'not self-starting'?

They require several previous points $y_{0}, y_{1}, \dots, y_{k-1}$ before they can begin, so a single-step method (e.g. RK4) must generate these starting values.

What is the purpose of adaptive step-size control in ODE solvers?

To automatically vary $h$ so the estimated local error stays within a user tolerance — using small steps where the solution changes rapidly and large steps where it is smooth, balancing accuracy and efficiency.

How does the Runge-Kutta-Fehlberg (RKF45) method estimate local error?

It computes both a 4th-order and a 5th-order RK estimate sharing the same function evaluations; their difference approximates the local truncation error used to accept/reject a step and adjust $h$.

Given an estimated error $E$, tolerance $\text{tol}$, and current step $h$ for a method of order $p$, what formula rescales the step?

$$h_{\text{new}} = h \left(\frac{\text{tol}}{E}\right)^{1/(p+1)}$$ often multiplied by a safety factor (e.g. $0.9$).

What is embedding in the context of adaptive Runge-Kutta methods?

An embedded RK pair shares stage evaluations $k_{i}$ between two methods of different order (e.g. orders 4 and 5), giving a nearly free error estimate from the difference of the two solutions.

What is a stiff differential equation?

An ODE (or system) with widely separated time scales — some solution components decay much faster than others — so explicit methods require impractically small step sizes for stability even when the solution is smooth.

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).

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.

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 $\operatorname{Re}(\lambda)<0$.

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).

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.

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.

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 $\ge 1$).

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.

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 $h\to 0$.

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 $p$.

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 $k+2$).

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.

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 $h$.

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: $y(x_{0}), y'(x_{0}), \dots, y^{(n-1)}(x_{0})$.

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.

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.

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 $$s = s_{0} + (s_{1}-s_{0})\frac{\beta - y(b;s_{0})}{y(b;s_{1}) - y(b;s_{0})}$$

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.

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 $O(h^{2})$.

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.

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

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).

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 $p$.

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.