Differential Equations

The Geometric View of ODEs

01Direction Fields and the Geometric View of ODEs 02Integral Curves and the Isocline Method 03Perpendicular Isoclines: The Circles of y' = -x/y 04Isocline Corridors and a Trap for Solutions 05Existence and Uniqueness: One Curve Through a Point 06Where Existence and Uniqueness Fail Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Numerical Methods

01Euler's Method and Its Three Equations 02Euler's Method by Hand: Building the Table 03Convexity and the Sign of Euler's Error 04First Order Error: Halve the Step, Halve the Error 05Improved Euler (RK2): Averaging Two Slopes 06Runge-Kutta 4 and the Price of Accuracy 07Singular Points: Where Every Method Fails Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

First-Order Linear Equations

01First Order Linear: Standard Form and the Sign of p 02The Conduction-Diffusion Model: Newton's Law of Cooling 03The Integrating Factor and the Four-Step Method 04Integrating Factors: Two Mistakes and a Trick Question 05Constant Coefficients: Steady State and Transient Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Improved Euler (RK2): Averaging Two Slopes

Why does averaging the slope where you stand with the slope at the end of the strut drop the error from $Ch$ to $C_2h^2$?

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Your summary note

1. Polling the direction field twice
  Describe taking the Euler slope $A_n = f(x_n, y_n)$ where you stand and the slope $B_n$ at the endpoint of the Euler strut, then stepping with their average $(A_n + B_n)/2$ .
2. The equations of the method
  Write $x_{n+1} = x_n + h$ , the temporary value $\tilde y_{n+1} = y_n + h A_n$ with $B_n = f(x_{n+1}, \tilde y_{n+1})$ , and the final $y_{n+1} = y_n + h\frac{A_n + B_n}{2}$ , marking $\tilde y_{n+1}$ as temporary.
3. The four names
  Record that Heun, Improved Euler, Modified Euler, and RK2 all name the same method, and that RK stands for Runge–Kutta.
4. Second order accuracy $\text{error} \approx C_2 h^2$
  State that the method is second order with error scaling like $C_2 h^2$ , and note that halving $h$ quarters the error.
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