GeometricIntegrators.jl

Julia library of geometric integrators for differential equations.

GeometricIntegrators.jl is a library of geometric integrators for ordinary differential equations, stochastic differential equations and differential algebraic equations in Julia. Its main purpose is the democratization and proliferation of geometric integrators by providing a comprehensive collection of structure-preserving as well as standard algorithms under a unified interface. GeometricIntegrators.jl can be used either interactively, as computational core in other codes, or from within DifferentialEquations.jl via GeometricIntegratorsDiffEq.jl. It provides both, a high-level interface that requires only very few lines of code to solve an actual problem, and a lean low-level interface that allows for straightforward integration into application codes via the exchange of minimalistic data structures. In both, the library leaves maximum control to the user. While trying to pick sensible defaults, all settings are accessible to and modifiable by the user. Suitable abstraction layers allow to choose between different linear and nonlinear solvers, auto-differentiation packages or custom routines for the computation of Jacobians and the like.

Package Description

Differential equations are ubiquitous in science and engineering. Many equations possess geometric features or abstract mathematical structures that need to be preserved in the discretisation in order to obtain reliable simulation results, especially for nonlinear problems and long-time simulations. The preservation of such properties improves stability, bounds global error growth and reduces numerical artefacts. Robust, performant and structure-preserving solvers for different types of differential equations are thus needed across many disciplines. GeometricIntegrators.jl provides such solvers and makes them available for both direct use as well as integration into other codes. Furthermore, the implemented algorithms can also be used within the DifferentialEquations.jl ecosystem [[1]], which is the defacto standard differential equation solver for the Julia programming language [[2]].

GeometricIntegrators.jl provides a comprehensive library of geometric integration algorithms as well as some non-geometric algorithms. It collects native Julia implementations of many known methods under a unified interface. Once a problem is implemented in the GeometricIntegators framework, all of its algorithms can immediately be applied and their performance evaluated. This facilitates numerical experiments with a wide variety of algorithms, simplifies benchmarking, and makes it easy to identify the best algorithm for a given problem. The implemented algorithms include explicit, implicit and partitioned Runge-Kutta methods, SPARK methods, splitting methods, symplectic methods and variational integrators. Most methods are implemented in an abstract way that allows for the flexible choice of tableaus, approximation spaces, basis functions, quadrature rules, and thus order of convergence. It also means that adding e.g. a new Runge-Kutta or splitting method merely amounts to adding its tableau. Less standard algorithms can be added by extending the package with custom integrators.

As most geometric integrators are not easily combined with time step adaption in a structure-preserving way, GeometricIntegrators.jl does not provide any general infrastructure for adaptive time stepping. Nonetheless, individual integrators can implement their own adaptivity strategies as long as they provide a solution at a predefined, equidistant series of time steps.

One of the original aims of GeometricIntegrators.jl is to serve as a testbed for the development and analysis of novel algorithms. Due to the modular structure and the use of the multiple dispatch paradigm, the library can easily be extended, e.g., towards new algorithms or new types of equations. The library aims at providing efficient implementations of diverse algorithms in order to be able to perform simulations and benchmarks with millions or even billions of time steps that facilitate the study of the long-time behaviour of both numerical algorithms and dynamical systems. The current scope of applications is mainly small- to mid-size systems of differential equations, e.g., systems of ordinary differential equations or semidiscretisations of partial differential equations with a moderate number of degrees of freedom. It is envisaged that in the future GeometricIntegrators.jl will also be able to address larger problems, especially semidiscretisations of partial differential equations in higher dimensions. In particular, this requires interfaces to appropriate iterative and parallel linear solvers, which are still lacking.

Similar Software

A Julia package closely related to GeometricIntegrators.jl is DifferentialEquations.jl [[1]]. Although serving similar purposes, the scope of the two libraries is rather different. While DifferentialEquations.jl provides a feature-rich ecosystem for the solution of differential equations, the focus of GeometricIntegrators.jl is on algorithms. While DifferentialEquations.jl is based around adaptive time stepping algorithms, GeometricIntegrators.jl focuses on fixed time step methods, given that structure-preservation is not easily combined with time step adaptivity. Therefore both packages are rather complementary, and GeometricIntegrators.jl can in fact be used as backend for DifferentialEquations.jl via GeometricIntegratorsDiffEq.jl.

Manual

• Equation and Problem Types
• • Keyword Arguments
  • Ordinary Differential Equations (ODEs)
  • Differential Algebraic Equation (DAE)
  • Stochastic Differential Equations (SDEs)
• List of all Methods
• • ODE Methods
  • PODE Methods
  • HODE Methods
  • IODE Methods
  • LODE Methods
  • SODE Methods

Integrators

• Runge-Kutta Methods
• • Common Runge-Kutta Methods
  • Simplifying Assumptions
  • Gauß, Lobatto and Radau Methods
  • Partitioned Equations
  • Implicit Equations
  • Custom Tableaus
• Splitting and Composition Methods
• • Splitting Integrators
  • Composition Integrators
  • Splitting of Hamiltonian Systems
  • Separable Hamiltonian Systems
  • Fourth Order Methods
  • Higher Order Methods by Composition
  • Splitting Coefficients
• Variational Integrators
• • Discretisation of the Action
  • Discrete Euler-Lagrange Equations
  • Discrete Fibre Derivative
  • Discrete Symplectic Form
  • Discrete Noether Theorem
  • Available Variational Integrators
• Variational Partitioned Runge-Kutta Integrators
• • Discrete Action Princtiple
  • Lobatto Methods
  • Degenerate Lagrangians
  • Dirac Constraints
  • Projection Methods
• Special Partitioned Additive Runge-Kutta Integrators
• Degenerate Variational Integrators
• Continuous Galerkin Variational Integrators
• • Space of Discrete Trajectories
  • Piecewise Polynomials
  • Numerical Quadrature
  • Galerkin Variational Integrators
• Discontinuous Galerkin Variational Integrators
• • Degenerate Lagrangian Systems
  • Discrete Trajectories and Numerical Quadrature
  • Numerical Fluxes
  • Discrete Variational Principle
• Hamilton-Pontryagin Integrators
• • Trapezoidal Discrete Lagrangian
  • Midpoint Discrete Lagrangian
• Hamilton-Pontryagin-Galerkin Integrators

Modules

• Equations
• • Ordinary Differential Equations
  • Differential Algebraic Equations
  • Stochastic Differential Equations
• Integrators
• • Geometric Integrator
  • Initial Guesses
  • Extrapolation Methods
• Euler Integrators
• • Runge-Kutta Integrators
  • Variational Integrators
  • Degenerate Variational Integrators
  • Galerkin Variational Integrators
  • Hamilton-Pontryagin Integrators
  • Splitting and Composition Methods
• Methods
• Projections
• • Projection Methods
  • Projected VPRK Methods
• Problem Types
• • Geometric Equation Problems
  • Ordinary Differential Equations
  • Differential Algebraic Equations
  • Stochastic Differential Equations
• Runge-Kutta Tableaus
• • Ordinary Runge-Kutta Tableaus
  • Partitioned Runge-Kutta Tableaus
• Solutions
• Solution Steps
• SPARK

Developer Documentation

• Integrators
• Code Integration
• Custom Integrators
• Adaptive Time Stepping

References

If you use GeometricIntegrators.jl in your work, please consider citing it by

@misc{Kraus:2020:GeometricIntegrators,
  title={GeometricIntegrators.jl: Geometric Numerical Integration in Julia},
  author={Kraus, Michael},
  year={2020},
  howpublished={\url{https://github.com/JuliaGNI/GeometricIntegrators.jl}},
  doi={10.5281/zenodo.3648325}
}

GeometricIntegrators.jl contains reference implementation for the methods described in the following articles:

• Michael Kraus. Projected Variational Integrators for Degenerate Lagrangian Systems. arXiv:1708.07356.
• Michael Kraus and Tomasz M. Tyranowski. Variational Integrators for Stochastic Dissipative Hamiltonian Systems. arXiv:1909.07202, Journal.

References for most of the available Runge-Kutta methods can be found in the documentation of RungeKutta.jl.

Background Material

• Ernst Hairer and Christian Lubich. Numerical Solution of Ordinary Differential Equations. The Princeton Companion to Applied Mathematics, 293-305, 2015. Princeton University Press. (Author's Web Site)
• Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration Illustrated by the Störmer–Verlet Method. Acta Numerica 12, 399-450, 2003. (Journal)
• John C. Butcher. Gauss Methods. Encyclopedia of Applied and Computational Mathematics, Pages 583-589, 2015. (Article)
• Laurent O. Jay. Lobatto Methods. Encyclopedia of Applied and Computational Mathematics, Pages 817–826, 2015. (Article)
• Ernst Hairer and Gerhard Wanner. Radau Methods. Encyclopedia of Applied and Computational Mathematics, Pages 1213-1216, 2015. (Article)

Books on Geometric Numerical Integration

• Sergio Blanes, Fernando Casas. A Concise Introduction to Geometric Numerical Integration. CRC Press, 2016. (eBook)
• Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration. Springer, 2006. (eBook)
• Benedict Leimkuhler and Sebastian Reich. Simulating Hamiltonian Dynamics. Cambridge University Press, 2005. (eBook)
• Jesús Maria Sanz-Serna, Manuel P. Calvo. Numerical Hamiltonian Problems. Chapman Hall, 1994.

Books on the Numerical Integration of Differential Equations

• Ernst Hairer, Syvert P. Nørsett and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993. (eBook)
• Ernst Hairer and Gerhard Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1996. (eBook)
• Ernst Hairer, Christian Lubich, Michel Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer, 1989. (eBook)
• Peter Deuflhard, Folkmar Bornemann. Scientific Computing with Ordinary Differential Equations. Springer, 2002. (eBook)
• John C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, 2016. (eBook)