GeometricEquations

Documentation for GeometricEquations.

In GeometricEquations.jl we define three basic types of equations:

• ordinary differential equations (ODEs),
• differential algebraic equations (DAEs),
• stochastic differential equations (SDEs).

For each type, there are several subtypes

• standard equations (ODE, DAE, SDE),
• implicit equations (IODE, IDAE),
• partitioned equations (PODE, PDAE, PSDE),
• Hamiltonian equations (HODE, HDAE),
• Lagrangian equations (LODE, LDAE),
• split equations (SODE, SPSDE).

Each equation holds a number of functions determining the vector field, constraints, and possibly additional information like periodicity, invariants and the Hamiltonian or Lagrangian. In addition to each equation type, GeometricEquations.jl implements a corresponding problem type. Each problem holds an equation, a time span (t₀,t₁) to integrate over, a time step to be used in the simulation, initial conditions and optionally parameters.

• The EquationProblem type holds an equation together with initial conditions and parameters, the time span and the time step of a simulation.
• The EnsembleProblem type holds an equation together with several initial conditions and/or parameters, etc.

GeometricEquations used to be part of GeometricIntegrators and is primarily used to define equations and problems for GeometricIntegrators. GeometricProblems contains various predefined problems and EulerLagrange can be used to generate code for equations from action principles.