Numerical Methods for Physics Simulations

Auto audio on: the next questions will be read aloud when you click Continue.

1. 1

   Why can a finite-difference derivative become less accurate when its step size is made extremely small?

   A The derivative necessarily becomes discontinuous B Round-off and subtractive cancellation can dominate C The truncation error always increases as the step shrinks
2. 2

   The diagram shows function samples at x−h, x, and x+h. Which expression is the centered finite-difference approximation to f′(x)?

   

   A [f(x+h) − f(x−h)]/(2h) B [f(x+h) − f(x)]/h C [f(x+h) − 2f(x) + f(x−h)]/h²
3. 3

   A numerical path is constructed by repeatedly following the slope at the beginning of each time step. Which update rule is being illustrated?

   

   A yₙ₊₁ = yₙ + h f(tₙ₊₁, yₙ₊₁) B yₙ₊₁ = yₙ + h[f(tₙ,yₙ) + f(tₙ₊₁,yₙ₊₁)]/2 C yₙ₊₁ = yₙ + h f(tₙ, yₙ)
4. 4

   For y′ = −λy with λ > 0, when is the explicit Euler method stable?

   A hλ > 2 B 0 < hλ < 2 C For every positive value of hλ
5. 5

   How many evaluations of the derivative function are performed during one step of the classical fourth-order Runge–Kutta method?

   A Four B Two C Eight
6. 6

   A computed phase-space trajectory for an ideal harmonic oscillator spirals outward instead of remaining closed. What does this most directly indicate?

   

   A Physical damping in the oscillator B Exact numerical conservation of energy C A numerical method causing secular energy gain
7. 7

   Which formula represents the composite trapezoidal rule on equally spaced points x₀ through xₙ with spacing h?

   

   A h[f(x₀) + f(x₁) + ··· + f(xₙ₋₁)] B h[f(x₀) + 2Σᵢ₌₁ⁿ⁻¹f(xᵢ) + f(xₙ)]/2 C h[f(x₀) + 4Σᵢ₌₁ⁿ⁻¹f(xᵢ) + f(xₙ)]/3
8. 8

   For exact function values and equally spaced samples, Simpson's 1/3 rule integrates which polynomials exactly?

   A All polynomials of degree at most three B Only polynomials of degree at most one C All polynomials of degree at most five
9. 9

   How does the typical statistical error of a basic Monte Carlo estimate scale with the number N of independent samples?

   A As N⁻² B As N⁻¹ C As N⁻¹ᐟ²
10. 10

    For the standard explicit centered finite-difference scheme for the one-dimensional wave equation, which condition is required for stability?

    

    A cΔt/Δx ≥ 2 B cΔt/Δx ≤ 1 C Δt must equal Δx regardless of c
11. 11

    A heat-equation model holds the temperature at each edge of a rectangular plate at specified fixed values. What type of boundary condition is this?

    

    A Dirichlet boundary condition B Neumann boundary condition C Periodic boundary condition
12. 12

    Why are matrices produced by finite-difference discretizations of local differential operators usually sparse?

    A Most physical fields are exactly zero B Boundary conditions remove every off-diagonal entry C Each grid point is coupled to only a few nearby points
13. 13

    A second-order numerical method has leading error proportional to h². Approximately what happens to this error when h is halved?

    A It is halved B It is reduced by a factor of four C It is reduced by a factor of eight
14. 14

    Which sequence correctly describes one position-and-velocity update in the velocity Verlet method?

    A Update position using aₙ, calculate aₙ₊₁, then update velocity using the average of aₙ and aₙ₊₁ B Update position and velocity using only the initial velocity, without evaluating acceleration C Choose the next position randomly and estimate velocity from the sample mean
15. 15

    What is the typical computational complexity of a fast Fourier transform applied to N data points?

    A O(N²) B O(1) C O(N log N)
16. 16

    Why are implicit methods often preferred for stiff systems of differential equations?

    A They never require solving algebraic equations B They can remain stable at time steps that make explicit methods unstable C They automatically eliminate every discretization error