List of numerical analysis topics

Euler integration — the most basic method for solving an ODE
Explicit and implicit methods — implicit methods need to solve an equation at every step
Runge-Kutta methods — one of the two main classes of methods for initial value problems
- Midpoint method — a second-order method with two stages
Multistep method — the other main class of methods for initial value problems
Methods designed for the solution of ODEs from classical physics:
- Newmark-beta method — based on the extended mean value theorem
- Verlet integration — a popular second-order method
- Beeman's algorithm — a two-step method extending the Verlet method
Geometric integrator — a method that preserves some geometric structure of the equation
- Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure
Stiff equation — roughly, an ODE for which the unstable methods needs a very short step size, but stable methods do not.
Shooting method — a method for the solution of boundary value problems
- Illustration of the shooting method

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Numerical partial differential equations

Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)

Methods for the solution of PDEs:
- Finite difference method — based on approximating differential operators with difference operators
  - Discrete Laplace operator — finite-difference approximation of the Laplace operator
  - Crank-Nicolson method — second-order method for heat and related PDEs
- Finite element method, finite element analysis — based on a discretization of the space of solutions
  - Finite element method in structural mechanics — a physical approach to finite element methods
  - Galerkin method — a finite element method in which the residual is orthogonal to the finite element space
  - Rayleigh-Ritz method — a finite element method based on variational principles
  - Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
- Spectral method — based on the Fourier transformation
  - Pseudo-spectral method
- Boundary element method — based on transforming the PDE to an integral equation on the boundary of the domain
- Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically
- Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics
- Discrete element method — a method in which the elements can move freely relative to each other
- Uniform theory of diffraction — specifically designed for scatting problems in electromagnetics.
Techniques for improving these methods:
- Multigrid, multigrid method — uses a hierarchy of nested meshes to speed up the methods
- Domain decomposition method — divides the domain in a few subdomains and solves the PDE on these subdomains
- Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary
Broad classes of methods:
- Mimetic methods — methods that respect in some sense the structure of the original problem
- Multiphysics — models consisting of various submodels with different physics
Analysis:
- Courant–Friedrichs–Lewy condition — stability condition for hyperbolic PDEs
- Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present
- Numerical resistivity — the same, with resistivity instead of diffusion
- Weak formulation — a functional-analytic reformulation of the PDE necessary for some methods