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∞

Runge-Kutta 4 and the Price of Accuracy

If RK4 costs four function evaluations per step, why does its fourth-order accuracy buy you nothing RK2 cannot?

Loading…

←Previous Improved Euler (RK2): Averaging Two Slopes Next Singular Points: Where Every Method Fails →

Your summary note

1. Function evaluations set the cost
  State that running time is governed by how many times the slope function is evaluated per step, and record the counts: Euler one, RK2 two, RK4 four, with the count roughly giving the order.
2. Trading accuracy for work — nothing is free
  Write that RK4's error goes like $h^4$ so halving $h$ cuts error by sixteen, then show that quartering the step in RK2 buys the same sixteenth for the same fourfold work.
3. RK4 super slope $\frac{A_n + 2B_n + 2C_n + D_n}{6}$
  Record that RK4 is the standard method drawing the on-screen curves, write the weighted super slope formula, and note the division by six from $1+2+2+1=6$ before plugging into the $y_{n+1}$ update.
Attempt 1 of 2