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GeometricIntegrators.jl • Features

The following list provides and overview of supported and planned features.

Families of Standard Methods

  • Explicit Runge-Kutta Methods (ERK),
  • Diagonally Implicit Runge-Kutta Methods (DIRK),
  • Fully Implicit Runge-Kutta Methods (FIRK),
  • Gauss-Legendre Runge-Kutta Methods (GLRK),
  • Radau I, Radau II and Lobatto III Runge-Kutta Methods,
  • Explicit Partitioned Runge-Kutta Methods (EPRK),
  • Implicit Partitioned Runge-Kutta Methods (IPRK),
  • Additive Runge-Kutta Methods (ARK),
  • Partitioned Additive Runge-Kutta Methods (PARK),
  • Generalised Additive Runge-Kutta Methods (GARK),
  • Specialised Partitioned Additive Runge-Kutta Methods (SPARK),
  • Continuous-stage Runge-Kutta Methods (CSRK),
  • Two-step Runge-Kutta Methods (TSRK),
  • General Linear Methods (GLM).

Families of Geometric Integrators

  • Variational Partitioned Runge-Kutta Methods (VPRK),
  • Hamiltonian Partitioned Additive Runge-Kutta Methods (HPARK, HSPARK),
  • Variational Partitioned Additive Runge-Kutta Methods (VPARK, VSPARK),
  • Continuous Galerkin Variational Integrators (CGVI),
  • Discontinuous Galerkin Variational Integrators (DGVI),
  • Hamilton-Pontryagin-Galerkin Integrators (HPGI),
  • Spline Variational Integrators (SVI),
  • Taylor Variational Integrators (TVI),
  • Degenerate Variational Integrators (DVI),
  • Splitting Methods (SM),
  • Hamiltonian Boundary Value Methods (HBVM).

Families of Stochastic Integrators

  • Stochastic Explicit Runge-Kutta Methods (SERK),
  • Stochastic Implicit Runge-Kutta Methods (SIRK),
  • Stochastic Implicit Partitioned Runge-Kutta Methods (SIPRK),
  • Stochastic Implicit Partitioned Additive Runge-Kutta Methods,
  • Stochastic Weak Explicit Runge-Kutta Methods (WERK),
  • Stochastic Weak Implicit Runge-Kutta Methods (WIRK).

Families of Equations

  • Systems of ODEs,
  • Systems of DAEs,
  • Systems of SDEs,
  • Partitioned ODEs,
  • Partitioned DAEs,
  • Partitioned SDEs,
  • Implicit ODEs,
  • Implicit DAEs,
  • Implicit SDEs,
  • Variational ODEs,
  • Hamiltonian DAEs,
  • Split ODEs,
  • Split Partitioned ODEs,
  • Split Partitioned SDEs.

Linear Solvers

  • LU decomposition (LAPACK),
  • LU decomposition (native Julia),
  • Krylov,

Nonlinear Solvers

  • Fixed-Point Iteration,
  • Fixed-Point Iteration with Aitken Acceleration,
  • Fixed-Point Iteration with Anderson Acceleration,
  • Jacobian-free Newton-Krylov,
  • Newton's method,
  • Newton's method with line search (Armijo, quadratic),
  • Quasi-Newton,

with

  • Analytic Jacobian,
  • Finite Difference Jacobian,
  • Jacobian obtained via Automatic Differentiation.

Diagnostics

  • Symplecticity Conditions,
  • Runge-Kutta Stability Area,
  • Convergence Analysis,
  • First Poincaré Integral Invariant,
  • Second Poincaré Integral Invariant.