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
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Math
// 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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