Better Numerical Methods

Sampling the slope more than once per step.

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

What this lesson teaches

The student compares numerical methods by accuracy and cost.

Euler commits to the starting slope for the whole step, which biases every step the same way.
Improved Euler averages two slopes; Runge-Kutta weights four.
Higher order means error falling far faster as the step shrinks, which is worth the extra work.

Straightforward practice. Get the method working first.

5 problems

Slopes of 5 and 9 at the ends of a step. What is their average?
Answer 7
Why 7.
Why do better methods sample the slope more than once per step?
Answer The slope changes during the step, so one sample at the start is biased.
Why The starting slope is not representative.
Euler's method. How many slope evaluations per step?
Answer 1
Why Just the one at the start.
Improved Euler. How many slope evaluations per step?
Answer 2
Why Two, then averaged.
Slopes of 2 and 6. What is their average?
Answer 4
Why Add and halve.

The same ideas with more to keep track of.

3 problems

What is the order of Euler's method?
Answer 1
Why First order.
What is the order of the standard Runge-Kutta method?
Answer 4
Why Fourth order.
A fourth-order method with the step halved. By roughly what factor does the error fall?
Answer 16
Why 2 to the fourth.

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

4 problems

Sort each method by its order.
Answer First or second order: Euler's method, Improved Euler · Fourth order: Standard Runge-Kutta, The classic four-stage method
Why Two of these are the same method under different names.
A second-order method with the step halved. By roughly what factor does the error fall?
Answer 4
Why 2 squared.
The Average: Slopes of 3 and 7 at the two ends of a step. What is their average?
Answer 5
Why 5.
The Cost: Standard Runge-Kutta. How many slope evaluations does it use per step?
Answer 4
Why 4.