final exam associated questions

MTH3007b Final Exam - Question Types

Related: Revision Overview / Cheat Sheet

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Based on the Chapra & Canale exercise list from the PDF (session 11) and the module content, the following question types can be expected.

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Block A: ODEs

1st Order IVPs

• Implement explicit Euler for a given ODE; report $y(t_{\max })$ for two step sizes.
• Implement implicit Euler for a given ODE; derive the closed-form rearrangement algebraically.
• Implement Ralston’s method; verify second-order convergence from error ratios.
• Implement RK4; compare accuracy with Euler for the same step size.
• Identify which method is appropriate for a stiff ODE; explain why explicit Euler fails.

Error and Order

• Given error at $dt=h$, predict error at $dt=h/2$ for a method of specified order.
• Compute error ratios from a table of numerical results; infer the order of the method.
• State the LTE and GTE of a given method; explain the relationship between them.

Algebraic Isolation

• Rearrange a linear or nonlinear equation to isolate $y$ or $y_{n+1}$ (including quadratic cases).
• Derive the implicit Euler update for a given linear ODE $\dot{y}=f(t,y)$.
• Derive the implicit trapezoid update for a given linear ODE.

Stability

• State the stability condition for explicit Euler applied to $\dot{y}=-ay$.
• Derive or state the amplification factor $G$ for explicit/implicit Euler.
• Explain the Lax Equivalence Theorem and what it guarantees.
• State the definitions of: LTE, GTE, consistency, stability, convergence, order.

Systems and Reductions

• Reduce a 2nd-order ODE to a 1st-order system; implement numerically.
• Implement a coupled 2D system (e.g. predator-prey); interpret the phase plane.
• Solve a spring-mass or coupled-compartment system using RK4.

Numerical Integration

• Compute a definite integral by treating it as an ODE; implement forward Euler or RK4.
• Compare numerical integration result with the analytical value.

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Block B: PDEs

FTCS (Heat Equation)

• Implement FTCS for the 1D heat equation with given parameters; check $r\le 1/2$.
• Compute the maximum stable $dt$ given $\alpha$ and $dx$.
• Demonstrate instability when $r>1/2$; describe what happens to the numerical solution.

BTCS (Heat Equation)

• Set up the tridiagonal matrix $A$ for BTCS; implement the solve using np.linalg.solve.
• Verify that BTCS remains stable for $r\gg 1/2$.
• Explain why BTCS is unconditionally stable (amplification factor argument).

Boundary Conditions

• Implement Dirichlet BCs in both FTCS and BTCS.
• Implement Neumann ($du/dx=0$) BCs using the imaginary point method; modify the matrix $A$ accordingly.
• Describe the physical meaning of a Neumann BC (insulated boundary).

Richardson Method

• State the Richardson (symmetric) scheme and its formal accuracy.
• Explain why it is unconditionally unstable despite higher formal order.

2D Laplace / Liebmann

• Implement Liebmann’s method for a given 2D plate with Dirichlet BCs.
• Derive the finite difference discretisation of the 2D Laplace equation.
• Interpret the stopping criterion $\max |u^{\text{new}}-u^{\text{old}}|<\epsilon$.

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Block C: Stochastic

Monte Carlo Integration

• Estimate a 1D integral using Monte Carlo with $N$ samples; report estimate and standard error.
• State and verify the $O(N^{-1/2})$ error scaling.
• Explain why MC is advantageous for high-dimensional integrals.

Wiener Process

• Implement a Wiener process simulation; plot multiple realisations.
• Verify that $\text{Var}[W(t)]=t$ from an ensemble.

OU Process and First-Passage Time

• Implement the OU process using Euler-Maruyama; identify mean-reverting behaviour.
• Find the first-passage time to a threshold for a single realisation.
• Estimate the mean first-passage time over many walkers; choose $N$ for 2 significant figures.
• Compute the standard error and 95% confidence interval for $⟨\tau ⟩$.

General Stochastic

• Write down the Euler-Maruyama update for a given Langevin SDE.
• Distinguish between a deterministic ODE and an SDE; identify the noise term.

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Likely Exam Format (Based on Module)

• Implement a given ODE solver from scratch (no framework code provided).
• Derive an algebraic rearrangement for an implicit method.
• Modify working code to change BCs or switch scheme.
• Interpret numerical output: read a table of errors and state the convergence order.
• Conceptual short answers: define a term, state a stability condition, explain a theorem.