Chapter 24 Quiz — PDEs and Finite Differences

Twenty questions to check that the stencil, the explicit step, the CFL condition, and the boundary conditions are solid before you move on. Aim for 16/20. Anything below that, revisit the section named in the "Topics to review" map at the end. Answers and one-line rationales are in the key.

Multiple choice

Q1. What distinguishes a partial differential equation from an ordinary one? - (a) It has more than one solution. - (b) Its unknown depends on several variables, so it involves partial derivatives. - (c) It cannot be solved on a computer. - (d) It is always nonlinear.

Q2. The five-point stencil approximates which operator? - (a) The gradient $\nabla u$. - (b) The first time derivative $\partial u/\partial t$. - (c) The Laplacian $\nabla^2 u$. - (d) The divergence $\nabla\cdot\mathbf{v}$.

Q3. In the square-grid five-point stencil $(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j})/h^2$, what is the role of the $1/h^2$? - (a) Cosmetic; it can be dropped. - (b) It converts a neighbour comparison into an actual second derivative. - (c) It enforces the boundary conditions. - (d) It guarantees stability.

Q4. A scheme is explicit when: - (a) the new values are given by a formula in already-known values only. - (b) it requires solving a linear system each step. - (c) it is unconditionally stable. - (d) it uses implicit none.

Q5. The 2D explicit (FTCS) heat scheme is stable when: - (a) $\alpha\Delta t/h^2 \le 1$. - (b) $\alpha\Delta t/h^2 \le 1/2$. - (c) $\alpha\Delta t/h^2 \le 1/4$. - (d) always — the heat equation is unconditionally stable.

Q6. You halve the grid spacing $h$ but keep $\alpha$ and $\Delta t$ fixed in an explicit 2D heat run. The diffusion number $r$: - (a) halves. - (b) is unchanged. - (c) doubles. - (d) quadruples.

Q7. A Dirichlet boundary condition fixes the boundary's: - (a) value. - (b) gradient (normal derivative). - (c) second derivative. - (d) time derivative.

Q8. A zero-Neumann edge (u(1) = u(2)) physically means: - (a) the edge is held at a fixed temperature. - (b) the edge is insulated — no heat crosses it. - (c) the domain wraps around at that edge. - (d) the edge temperature is undefined.

Q9. Which scheme is typically unconditionally stable (no timestep limit)? - (a) FTCS explicit. - (b) An implicit (e.g. backward-Euler / Crank–Nicolson) step. - (c) The five-point stencil. - (d) None; all explicit schemes have limits.

Q10. On a structured grid, a point's neighbours are found by: - (a) looking them up in a stored connectivity list. - (b) simple index arithmetic, $(i\pm1,j)$ and $(i,j\pm1)$. - (c) a nearest-neighbour search. - (d) solving a linear system.

True/False (justify in one line)

Q11. True or false: The heat equation and the diffusion equation are the same mathematics with $u$ reinterpreted.

Q12. True or false: Running an explicit heat solver exactly at $r = 1/4$ is perfectly safe and recommended.

Q13. True or false: For the wave equation, the CFL condition is the Courant number $C = c\Delta t/h \le 1$, meaning a wave may cross at most one grid cell per timestep.

Q14. True or false: You may safely overwrite u in place during the FTCS sweep, because the stencil only reads nearby cells.

Q15. True or false: The explicit stability limit scales as $\Delta t \sim h$, so halving $h$ only doubles the number of steps.

Short answer

Q16. In one sentence, what does the diffusion number $r = \alpha\Delta t/h^2$ collect, and why does it control both accuracy and stability?

Q17. Why must the FTCS update be computed from a snapshot of the old field (two buffers, or a fresh Laplacian array) rather than in place?

Q18. Name the two error sources in the FTCS heat solver and their orders (time and space).

What does this code print?

Q19. A $5$-node 1D rod, ends u(1)=0, u(5)=100, interior 0, one FTCS step with r = 0.25. What is u(4) after the step?

u_new(4) = u(4) + 0.25_dp*(u(5) - 2.0_dp*u(4) + u(3))

Q20. With the 2D stencil and r = 0.5, a checkerboard interior cell starts at 1.0 with two interior neighbours at -1.0 and two boundary neighbours at 0.0. What is its value after one step, and what does the sign-and-magnitude change tell you?

u_new(2,2) = u(2,2) + 0.5_dp*(u(1,2)+u(3,2)+u(2,1)+u(2,3) - 4.0_dp*u(2,2))

Answer Key

Q Answer Rationale
1 b A PDE's unknown depends on several variables → partial derivatives w.r.t. each.
2 c The five-point stencil is the discrete Laplacian $\nabla^2 u$.
3 b Dividing by $h^2$ turns a neighbour comparison into a true second derivative (with units).
4 a Explicit = new value is a formula in known (current-step) values; no system to solve.
5 c 2D FTCS is stable iff $r = \alpha\Delta t/h^2 \le 1/4$.
6 d $r \propto 1/h^2$, so halving $h$ quadruples $r$.
7 a Dirichlet fixes the boundary value.
8 b Zero-Neumann = zero gradient = insulated, no heat flux.
9 b Implicit schemes are typically unconditionally stable.
10 b Structured grid → neighbours by index arithmetic, no stored connectivity.
11 True Identical equation; "heat" vs "diffusion" is just the interpretation of $u$.
12 False At $r=1/4$ the worst mode has $|G|=1$ — marginally stable; round-off/nonlinearity can push it over. Stay safely below.
13 True That is the original (hyperbolic) CFL condition.
14 False In-place overwrite mixes time levels — it silently becomes Gauss–Seidel, a different scheme.
15 False Diffusion limit scales as $\Delta t \sim h^2$; halving $h$ needs $4\times$ more steps.
16 — It collects physics ($\alpha$), timestep ($\Delta t$), and grid ($h$); the update multiplies the stencil by $r$, so it sets both the step size and the amplification factor.
17 — FTCS is defined on the old neighbours $u^n$; in-place updates feed new values into later points, changing the scheme.
18 — Time: $O(\Delta t)$ (forward-Euler, first order). Space: $O(h^2)$ (central second difference).
19 25.0 $0 + 0.25(100 - 0 + 0) = 25$.
20 -2.0 $1 + 0.5(0 - 1 + 0 - 1 - 4) = 1 - 3 = -2$: sign flips and magnitude grows → unstable, $r=0.5 > 1/4$.

Topics to review by question

Questions Section to review
1, 11 §24.1 (heat/wave equations, PDEs)
2, 3 §24.2 (five-point stencil, the $1/h^2$ scaling)
4, 9, 16, 17, 18, 19 §24.3 (explicit vs implicit, FTCS, snapshot update)
5, 6, 12, 13, 15, 20 §24.4 (CFL / stability, the diffusion number)
7, 8, 14 §24.5 (boundary conditions)
10 §24.6 (structured grids)