Unit 5: Numerical Methods
What to do when no formula exists: stepping a solution forward numerically, and watching the error shrink as the steps get smaller.
6 topics · 21 questions
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8.1 Why Numerical Methods Matter
- How it worksHow does approximating a solution curve step by step get you close to an answer with no formula at all?
- Common mistakeWhy doesn't the lack of a closed-form solution mean a differential equation has no solution?
- Going furtherWhat happens to the usefulness of a numerical method when the true solution blows up in finite time?
- Worked exampleFind why the equation y' = e−x^2 has no elementary antiderivative to build a formula solution from.
Your turn1 of 2
The equation y' = 2 + y has no simple closed-form trick needed here — take one step of Euler's method with step size 0.395, starting from y(0) = 4, to estimate y(0.395).8.1 Euler's Method
- How it worksHow does Euler's method turn a differential equation into the update rule y sub n+1 equals y sub n plus h times f?
- How it worksHow does shrinking the step size h make Euler's method hug the true solution curve more closely?
- Going furtherWhat happens to an Euler's method approximation when the true solution curves sharply within a single step?
- Worked exampleFind the next two Euler's method points for y' = x - y, y(0) = 1, using step size 0.5.
Your turn1 of 2
For y' = x - y, y(0) = 3, using step size h = 0.395, find y1, the Euler's method estimate after one step.8.1 Stepping Through Euler's Method by Hand
- Worked exampleFind the Euler's method estimate of y(1) for y' = x + y, y(0) = 1, using step size 0.5.
- Worked exampleFind how much closer to the truth the y' = x + y, y(0) = 1 estimate lands using a smaller step size of 0.25 instead of 0.5.
- Common mistakeWhy doesn't taking twice as many steps in Euler's method cut the error exactly in half?
Your turn1 of 2
For y' = x + y, y(0) = 3, use two steps of Euler's method with step size h = 0.395 to estimate y(2h).8.1 Local and Global Error in Euler's Method
- How it worksHow do local truncation error and global error accumulate differently as Euler's method marches forward?
- Worked exampleCompare the Euler estimate of y(1) for y' = y, y(0) = 1 at step sizes 0.5, 0.25, and 0.125 against the exact value e.
- Going furtherWhat happens to Euler's method's global error as the step size h is repeatedly cut in half?
Your turn1 of 2
For y' = y, y(0) = 2, use Euler's method with step size h = 0.36 and n = 5 steps to estimate y(n·h). Find the estimate.8.2 The Improved Euler Method
- How it worksHow does the improved Euler method use a predictor step and a corrector step to average two slopes?
- Worked exampleFind the improved Euler method's estimate of y(1) for y' = 2x - y, y(0) = 1, using step size 0.5.
- Common mistakeWhy doesn't doubling the work per step in the improved Euler method only double its accuracy?
Your turn1 of 2
For y' = 2x - y, y(0) = 3, use one step of the improved Euler method with step size h = 0.395 to estimate y(h).8.2 Comparing Euler and Improved Euler Accuracy
- Worked exampleCompare the Euler and improved Euler estimates of y(1) for y' = -2y, y(0) = 1 at step size 0.25 against the exact solution.
- Going furtherWhat happens to the gap between Euler and improved Euler's accuracy as the step size shrinks toward zero?
- Common mistakeWhy doesn't a numerical method with a smaller step size automatically beat a different method using a larger one?
- Ties togetherHow does the order of a numerical method's error connect to how many terms of a Taylor series it's effectively matching?
Your turn1 of 2
For y' = -3y, y(0) = 3, compare one step of the improved Euler method to one step of plain Euler, both with step size h = 0.295. Find the difference: improved Euler's estimate minus plain Euler's estimate.