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GeometricIntegrators.jl

Features

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

Families of Standard Methods

  • [x] Explicit Runge-Kutta Methods (ERK),
  • [x] Diagonally Implicit Runge-Kutta Methods (DIRK),
  • [x] Fully Implicit Runge-Kutta Methods (FIRK),
  • [x] Explicit Partitioned Runge-Kutta Methods (EPRK),
  • [x] Implicit Partitioned Runge-Kutta Methods (IPRK),
  • [ ] Additive Runge-Kutta Methods (ARK),
  • [x] 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

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

Families of Stochastic Integrators

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

Families of Equations

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

Linear Solvers

  • [x] LU decomposition (LAPACK),
  • [x] 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,
  • [x] Newton's method,
  • [x] Newton's method with line search (Armijo, quadratic),
  • [x] Quasi-Newton,

with

  • [x] Analytic Jacobian,
  • [x] Finite Difference Jacobian,
  • [x] Jacobian obtained via Automatic Differentiation.

Diagnostics

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