Numerical Methods Summary

Differential Equations · Axiom Academy

A recap of how we approximate solutions when a differential equation resists exact methods — and how error analysis tells us which approximation to trust. Every numerical IVP method does the same thing at different orders of accuracy: sample the slope, step forward, repeat — the ORDER just changes how many slopes it samples and how the error shrinks. Euler's method is first-order (global error O(h) , 1 slope evaluation/step); Improved Euler/Heun is second-order ( O(h^2) , 2 evaluations/step); RK4 is fourth-order ( O(h^4) , 4 evaluations/step). Local truncation error (one step) is always one power of h higher than global error (accumulated over the whole interval) — that's true at every order. Boundary value problems can't march forward from one point — solve them by shooting (iterate an initial slope) or finite differences (discretize the whole domain at once). There's no universally "best" method: the right choice trades accuracy against computational cost, and adapts further for stiff equations or systems. Core Concept Why Numerical Methods? Most differential equations that show up in real modeling have no closed-form solution — so we approximate the solution curve numerically instead of solving for it exactly. When to use: whenever an analytical technique (separation, integrating factor, series) doesn't apply or is too costly to carry out Watch out for: smaller steps improve accuracy but cost more evaluations — every method is an accuracy/cost trade-off, not a free upgrade

This is the written version of the interactive lesson above. See the full Differential Equations course.