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Cogito

Differential Equations · Chapter 4 · Lesson 3

Practice Worksheet

Better Numerical Methods

Sampling the slope more than once per step.

12 problems · about 23 minutes · F-IF.B.6

NameDate

Figure — use these to answer the problems

This trajectory is stepped by Runge-Kutta and closes into a clean circle. Under Euler it would visibly spiral outward instead.
  • Released from(2, 0)
the rotation rate1
starting x2
starting y0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Slopes of 5 and 9 at the ends of a step. What is their average?

    Answer

  2. Why do better methods sample the slope more than once per step?

    • The slope changes during the step, so one sample at the start is biased.
    • To check the arithmetic twice.

  3. Euler's method. How many slope evaluations per step?

    Answer

  4. Improved Euler. How many slope evaluations per step?

    Answer

  5. Slopes of 2 and 6. What is their average?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. What is the order of Euler's method?

    Answer

  2. What is the order of the standard Runge-Kutta method?

    Answer

  3. A fourth-order method with the step halved. By roughly what factor does the error fall?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each method by its order.

    Write each item under the heading it belongs to: Euler's method · Improved Euler · Standard Runge-Kutta · The classic four-stage method

    First or second order

    Fourth order

  2. A second-order method with the step halved. By roughly what factor does the error fall?

    Answer

  3. The Average

    Slopes of 3 and 7 at the two ends of a step. What is their average?

    Answer

  4. The Cost

    Standard Runge-Kutta. How many slope evaluations does it use per step?

    Answer