Summary Numerical ODEs and Applications — Complete Module Revision Guide

Numerical ODEs and Applications — a rigorous applied mathematics/numerical analysis module covering the full theory of numerical methods for ordinary differential equations, from foundations through to advanced convergence theory. Written as a complete, restructured revision guide (not raw lecture notes) — organised by topic for efficient exam preparation, covering the full syllabus in 14 pages with full derivations and worked examples throughout. Index of topics covered: 1. Foundations and motivating examples (population models, Picard's theorem) 2. Euler's method and Taylor series methods (with full convergence proof) 3. General theory: consistency, stability & convergence 4. Quadrature, collocation & Runge-Kutta methods (deriving the RK order conditions) 5. Stability of Runge-Kutta methods (spurious fixed points, A-stability) 6. Linear multistep methods (Adams-Bashforth, BDF methods) 7. Order, convergence & the Dahlquist barriers This guide includes full mathematical derivations (not just definitions) for the key theorems of the course — including the complete convergence proof for Euler's method, the derivation showing collocation methods produce the general Runge-Kutta form, the classical RK order conditions, and both Dahlquist barriers governing the maximum achievable order and A-stability of linear multistep methods. Suitable for students studying numerical analysis, numerical methods for differential equations, or applied/computational mathematics modules with similar content.