I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization.
I'm happy to be part of the first Observable Ambassador's cohort.
Published
Edited
Nov 25, 2020
29 stars
// Perform Euler's method with n steps over the interval [a,b]
// for y' = f(y,t) with y(t0) = y0.
function euler(f, y0, a, b, n) {
let dt = (b - a) / n;
let ti = a;
let yi = y0;
let pts = [[ti, yi]];
for (let i = 0; i < n; i++) {
yi = yi + f(yi, ti) * dt;
ti = ti + dt;
pts.push([ti, yi]);
}
return pts;
}
pts = euler((y, t) => y, 1, 0, 1, 10)
// Apply the Runge-Kutta method with n steps over an interval
// [a,b] containing t0.
function rk4_1d(f, y0, t0, a, b, n) {
let dt = (b - a) / n;
if (t0 == a) { // Just go forward
return rk4_multi_step_1d(f, y0, a, dt, n);
} else if (b == t0) { // Just go backward
let result = rk4_multi_step_1d(f, y0, b, -dt, n);
result.reverse();
return result;
} else if (a < t0 && t0 < b) { // Go forward and backward and concatenate
let i = Math.floor((t0 - a) / dt);
let backward = rk4_multi_step_1d(f, y0, t0, -dt, i);
backward.reverse();
let forward = rk4_multi_step_1d(f, y0, t0, dt, n - i).slice(1);
return backward.concat(forward);
}
}
function rk4_multi_step_1d(f, y0, t0, dt, n) {
let result = [[t0, y0]];
for (let i = 0; i < n; i++) {
y0 = rk4_single_step_1d(f, y0, t0, dt);
t0 = t0 + dt;
if (Math.abs(y0) < 1000) {
result.push([t0, y0]);
} else {
break;
}
}
return result;
}
function rk4_single_step_1d(f, y0, t0, dt) {
let dt2 = dt / 2;
let k1 = f(y0, t0);
let k2 = f(y0 + k1 * dt2, t0 + dt2);
let k3 = f(y0 + k2 * dt2, t0 + dt2);
let k4 = f(y0 + k3 * dt, t0 + dt);
return y0 + ((k1 + 2 * k2 + 2 * k3 + k4) * dt) / 6;
}
md`Euler's method with 10 steps:`
euler_approx10 = euler(y => y, 1, 0, 1, 10).slice(-1)[0][1]
rk4_approx = rk4_1d(y => y, 1, 0, 0, 1, 10).slice(-1)[0][1]
euler_approx100000 = euler(y => y, 1, 0, 1, 100000).slice(-1)[0][1]
[Math.E - euler_approx100000, Math.E - rk4_approx]
// d3 = require('d3@6')
d3 = require('d3-selection@2', 'd3-scale@3', 'd3-shape@2', 'd3-array@2', 'd3-axis@2')
math = require('mathjs@7.6.0')
import { slider, text, select } from "@jashkenas/inputs"
import { plotter } from '@kjerandp/plotter'
// Store the values of t0 and y0 in this external object so
// they don't get rewritten when we resize R.
inits = ({
t0: null,
y0: null
})
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