List of Numerical Analysis Topics - Numerical Methods For Ordinary Differential Equations

Numerical Methods For Ordinary Differential Equations

Numerical methods for ordinary differential equations — the numerical solution of ordinary differential equations (ODEs)

• Euler method — the most basic method for solving an ODE
• Explicit and implicit methods — implicit methods need to solve an equation at every step
• Backward Euler method — implicit variant of the Euler method
• Trapezoidal rule — second-order implicit method
• Runge–Kutta methods — one of the two main classes of methods for initial-value problems
  • Midpoint method — a second-order method with two stages
  • Heun's method — either a second-order method with two stages, or a third-order method with three stages
  • Bogacki–Shampine method — a third-order method with four stages (FSAL) and an embedded fourth-order method
  • Cash–Karp method — a fifth-order method with six stages and an embedded fourth-order method
  • Dormand–Prince method — a fifth-order method with seven stages (FSAL) and an embedded fourth-order method
  • Runge–Kutta–Fehlberg method — a fifth-order method with six stages and an embedded fourth-order method
  • Gauss–Legendre method — family of A-stable method with optimal order based on Gaussian quadrature
  • Butcher group — algebraic formalism involving rooted trees for analysing Runge–Kutta methods
  • List of Runge–Kutta methods
• Linear multistep method — the other main class of methods for initial-value problems
  • Backward differentiation formula — implicit methods of order 2 to 6; especially suitable for stiff equations
  • Numerov's method — fourth-order method for equations of the form
  • Predictor–corrector method — uses one method to approximate solution and another one to increase accuracy
• Bulirsch–Stoer algorithm — combines the midpoint method with Richardson extrapolation to attain arbitrary order
• 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
  • Leapfrog integration — another name for Verlet integration
  • Beeman's algorithm — a two-step method extending the Verlet method
  • Dynamic relaxation
• 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
    • Variational integrator — symplectic integrators derived using the underlying variational principle
    • Semi-implicit Euler method — variant of Euler method which is symplectic when applied to separable Hamiltonians
  • Energy drift — phenomenon that energy, which should be conserved, drifts away due to numerical errors
• Other methods for initial value problems (IVPs):
  • Bi-directional delay line
  • Partial element equivalent circuit
• Methods for solving two-point boundary value problems (BVPs):
  • Shooting method
  • Direct multiple shooting method — divides interval in several subintervals and applies the shooting method on each subinterval
• Methods for solving differential-algebraic equations (DAEs), i.e., ODEs with constraints:
  • Constraint algorithm — for solving Newton's equations with constraints
  • Pantelides algorithm — for reducing the index of a DEA
• Methods for solving stochastic differential equations (SDEs):
  • Euler–Maruyama method — generalization of the Euler method for SDEs
  • Milstein method — a method with strong order one
  • Runge–Kutta method (SDE) — generalization of the family of Runge–Kutta methods for SDEs
• Methods for solving integral equations:
  • Nyström method — replaces the integral with a quadrature rule
• Analysis:
  • Truncation error (numerical integration) — local and global truncation errors, and their relationships
    • Lady Windermere's Fan (mathematics) — telescopic identity relating local and global truncation errors
• Stiff equation — roughly, an ODE for which the unstable methods needs a very short step size, but stable methods do not
  • L-stability — method is A-stable and stability function vanishes at infinity
  • Dynamic errors of numerical methods of ODE discretization — logarithm of stability function
• Adaptive stepsize — automatically changing the step size when that seems advantageous
• History of numerical solution of differential equations using computers