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.
Public
Edited
Nov 17, 2022
Importers
9 stars
A Julia set on the Riemann sphereThe Z-CurveBarnsley's fernA stochastic digraph IFS algorithmSelf-affine tilesThe TwindragonThe Eisenstein fractionsA self-affine tile with holesSelf-affine tiles via polygon mergeGolden rectangle fractalsBifurcation diagram with critical curvesThe tame twindragonIllustrations for the proof of Green's theoremNon-orientability of a Mobius stripExamples of parametric surfacesPenrose tilingThe extended unit circlePenrose three coloringNewtons's method on the Riemann sphereConic sectionsDivisor graphsThe dance of Earth and VenusIterating multiples of the sine functionBorderline fractalsSelf-similar intersectionsBox-counting dimension examplesMandelbrot by dimensionInverse iteration for quadratic Julia setsInteger Apollonian PackingsIllustrations of two-dimensonal heat flowThe logistic bifurcation locusThe eleven unfoldings of the cubeA unimodal function with fractal level curvesGreen's theorem and polygonal areaThe geometry and numerics of first order ODEsThe xxx^xxx-spindleAnimated beatsRauzy FractalsHilbert's coordinate functionsPluckNot PiDrum strikeThe Koch snowflakeFractalized squareA Taylor series about π/4\pi/4π/4PlotX3D HyperboloidA PlotX3D animationModular arithmetic in 5th grade artSimple S-I-R ModelThe Poisson KernelPoly-gasketsClassification of 2D linear systems via trace and determinantJulia sets and the Mandelbrot setWater wavesFourier SeriesDisks for a solid of revolutionOrbit detection for the Mandelbrot setTracing a path on a spherePlot for mathematiciansFunctions of two variablesPartial derivativesDijkstra's algorithm on an RGGGradient ascentUnfolding polyhedraTangent plane to a level surfaceA strange discontinuityExamples of level surfacesMcMullen carpetsHills and valleysThe definition of ⇒Double and iterated integralsMST in an RGGTrees are bipartiteFractal typesettingd3.hierarchy and d3.treeK23 is PlanarPolar CoordinatesParametric region generatorParametric Plot 2DContour plotsGreedy graph coloringGraph6A few hundred interesting graphsThe Kings ProblemFirst order, autonomous systems of ODEs
Runge-Kutta for systems of ODEs
Also listed in…
Teaching ODEs
orbit = rk4(([x, y], t) => [-y - x / t, x - y / 2], [1, 1], 1, 0.5, 100, 1000)
{
let w = 0.8 * width;
let h = 0.6 * w;
let plot = plotter({
xDomain: [-1, 3],
yDomain: [1.5, -0.5],
width: w,
height: h
});
plot.polyline(orbit.map(([x, y]) => ({ x: x, y: y })));
return plot.node;
}
// Run RK4 forwards and backwards over the time interval [a,b] where t0 is in [a,b], like
// rk4(([x]) => [x], [1], 0, -5, 5, 100)
function rk4(f, y0, t0, a, b, n) {
let dt = (b - a) / n;
if (t0 == a) {
// Just go forward
return rk4_multi_step(f, y0, a, dt, n);
} else if (b == t0) {
// Just go backward
let result = rk4_multi_step(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(f, y0, t0, -dt, i);
backward.reverse();
let forward = rk4_multi_step(f, y0, t0, dt, n - i).slice(1);
return backward.concat(forward);
}
}
// Take a bunch of RK4 steps, like
// rk4_multi_step(([x]) => [x], [1], 0, 0.1, 10)
function rk4_multi_step(f, y0, t0, dt, n) {
let y = y0;
y.t = t0;
let ys = [y];
for (let i = 0; i < n; i++) {
y = rk4_single_step(f, y, t0 + i * dt, dt);
y.t = t0 + (i + 1) * dt;
// Implement an overly simple bailout
// ys.push(y);
if (d3.sum(y.map((x) => x ** 2)) < 10 ** 8 && y.filter(isNaN).length == 0) {
ys.push(y);
} else {
return ys;
}
}
return ys;
}
// Take a single RK4 step from t0 to t0+dt, like
// rk4_single_step(([x]) => [x], [1], 0, 0.1)
function rk4_single_step(f, y0, t0, dt) {
let dt2 = dt / 2;
let k1 = f(y0, t0);
let k2 = f(add(y0, mul(dt2, k1)), t0 + dt2);
let k3 = f(add(y0, mul(dt2, k2)), t0 + dt2);
let k4 = f(add(y0, mul(dt, k3)), t0 + dt);
let s1 = add(k1, mul(2, k2));
let s2 = add(s1, mul(2, k3));
let s3 = add(s2, k4);
return add(y0, mul(dt / 6, s3));
}
// Multiply a vector by a scalar, like
// mul(2,[3,4])
function mul(r, a) {
return a.map(x => r * x);
}
// add two vectors, like
// add([1, 2], [3, 4])
function add(a1, a2) {
return a1.map((a, i) => a + a2[i]);
}
d3 = require('d3-array@2')
// Fork of '@kjerandp/plotter'
import { plotter } from '50dadfdec01c15a8'
n = 4
init = d3.range(0, 1 + 1 / n, 1 / n).map(x => 4 * (x - x ** 2))
d3.range(0, 1 + 1 / n, 1 / n)
function F(u, t) {
let v = Array.from({ length: n });
v[0] = 0;
v[n] = 0;
for (let i = 1; i < n; i++) {
v[i] = n ** 2 * (u[i + 1] - 2 * u[i] + u[i - 1]);
}
return v;
}
// rk4_multi_step(f, y0, t0, dt, n)
// rk4(f, y0, t0, a, b, n)
// orbit = rk4(([x, y], t) => [-y - x / t, x - y / 2], [1, 1], 1, 0.5, 100, 1000)
result = rk4(F, init, 0, 0, 1, 10)
import { odeRK4 } from '@rreusser/integration'
odeRK4
init
yy = d3.range(n + 1)
yy
init
rresult = {
let y = d3.range(0, 1 + 1 / n, 1 / n).map(x => 4 * (x - x ** 2));
for (let i = 0; i < 1000; i++) {
odeRK4(y, y, f, 0.001);
}
return y;
}
function f(_, u) {
let v = Array.from({ length: n });
v[0] = 0;
v[n] = 0;
for (let i = 1; i < n; i++) {
v[i] = n ** 2 * (u[i + 1] - 2 * u[i] + u[i - 1]);
}
return v;
}
Purpose-built for displays of data
Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.
