Numerical Schemes for Evolution Equations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Turning change into steps

Mathematics · Differential Equations · ages 23-24
Name ______________________   Date ____________

  1. The explicit heat scheme forces tiny time steps because stability needs r below or at 1 over 2.

    Circle one:   True   False

  2. The scheme using a forward time difference and a centred space difference has what order of accuracy?

    • Second order in time, first in space
    • First order in time, second in space
    • First order in both
    • Second order in both

  3. A student says the explicit heat scheme forces tiny time steps because stability needs r <= 1/2. Is that right?

    Circle one:   True   False

  4. An explicit heat scheme uses time step 0.01 and space step 0.1 with diffusivity 1. What is the mesh ratio r?

    Answer: ______________

  5. A student says the backward Euler (implicit) scheme for the heat equation is stable for any step sizes. Is that right?

    Circle one:   True   False

  6. With space step 0.1 and diffusivity 1, what is the largest stable time step for the explicit heat scheme?

    • 0.005
    • 0.05
    • 0.5
    • 0.1

  7. One explicit step with r = 0.5 updates the value 4 from neighbours 1 and 2. What is the new value?

    Answer: ______________

  8. The scheme uses a forward time difference and a centred space difference. What is its order?

    • Second in time, first in space
    • First in time, second in space
    • Exact in both time and space

  9. Lee halves dx, keeps dt, and trusts the old stability check. What is wrong?

    • He used too small a dt
    • He blended too few neighbours
    • He changed r without rechecking the bound

  10. A student doubles the space step and quarters the time step, claiming r stays fixed. Is that right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Turning change into steps W1-mt_RuDbOapexB-s1

  1. True · The bound ties dt to the square of dx.
  2. First order in time, second in space · Forward time differencing is first order while centred space differencing is second, so first order in time, second in space.
  3. True · Stability demands r <= 1/2, which ties the time step to the square of the space step.
  4. 1 · r is dt over dx squared: 0.01 over 0.01 is 1.
  5. True · Backward Euler damps every mode regardless of step size, hence unconditional stability.
  6. 0.005 · Stability needs dt <= (dx^2)/2 = 0.005, so 0.005 is the largest safe step.
  7. 1.5 · The update gives 4 + 0.5 times (1 - 8 + 2) = 4 - 2.5 = 1.5.
  8. First in time, second in space · Forward time is first order; centred space is second.
  9. He changed r without rechecking the bound · The same dt on a finer grid inflates r past the bound.
  10. True · Doubling dx multiplies dx^2 by 4 while dt is quartered, so their ratio r is unchanged.
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