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Summary Numerical Partial Differential Equations — Complete Module Revision Guide

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2025/2026 Written for academic year
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The University Of Edinburgh (ED)
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Numerical Partial Differential Equations
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Numerical Partial Differential Equations — a rigorous applied mathematics/numerical analysis module covering the full theory and derivation of finite difference methods for PDEs. Written as a complete, restructured revision guide (not raw lecture notes) — organised by topic for efficient exam preparation, covering the full syllabus in 15 pages with full derivations and worked examples throughout. Index of topics covered: 1. Classification of PDEs (hyperbolic, elliptic, parabolic) 2. Finite differences for ODEs: consistency, stability & convergence 3. The method of lines and the heat equation (forward Euler, backward Euler, Crank-Nicolson) 4. Boundary conditions (Dirichlet, Neumann, mixed) 5. Finite differences for elliptic PDEs (Poisson equation, five-point Laplacian) 6. Spectral methods (non-examinable overview — Fourier & Chebyshev methods) 7. Iterative methods for linear systems (Jacobi, Gauss-Seidel, steepest descent, conjugate gradients) 8. Hyperbolic PDEs and the advection equation (characteristics, upwind schemes, Courant number) This guide includes full mathematical derivations (not just definitions) for the key theorems of the course — including the fundamental "consistency + stability = convergence" theorem, the stability condition for forward Euler applied to the heat equation, and the convergence rate of the conjugate gradient method. Suitable for students studying numerical analysis, numerical methods for PDEs, or applied/computational mathematics modules with similar content.

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Numerical Partial Differential
Equations
Complete Revision & Study Guide

A topic-by-topic guide covering PDE classification, finite difference methods, stability theory,
elliptic and parabolic problems, iterative linear solvers, and hyperbolic PDEs — with full
derivations and worked examples throughout.



Original study notes — independently written summary and explanation

, Contents

• 1. Classification of PDEs
• 2. Finite Differences for ODEs: Consistency, Stability & Convergence
• 3. The Method of Lines and the Heat Equation
• 4. Boundary Conditions
• 5. Finite Differences for Elliptic PDEs
• 6. Spectral Methods (non-examinable overview)
• 7. Iterative Methods for Linear Systems
• 8. Hyperbolic PDEs and the Advection Equation

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Subjects

  • numerical pdes
  • pde classification
  • finite difference methods
  • finite differences
  • consistency stability
  • convergence
  • lax richtmyer stability
  • von neumann analysis
  • numerical partial differential equations
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