Runge-Kutta Methods

This package collects Runge-Kutta tableaus and provides diagnostics to analyze them. It implements algorithms for the computation of Gauss, Lobatto and Radau tableaus with arbitrary numbers of stages as well as tabulated coefficients for various explicit, diagonally implicit and fully implicit methods. All tableaus can be retrieved in arbitrary precision or symbolically.

Installation

RungeKutta.jl and all of its dependencies can be installed via the Julia REPL by typing

]add RungeKutta

Basic Usage

After loading the Runge-Kutta module by

julia> using RungeKutta

a Tableau can be created by calling any one of the provided constructors, for example

julia> TableauExplicitMidpoint()

Runge-Kutta Tableau explicit_midpoint with 2 stages and order 2:

 0.0 │ 0.0  0.0
 0.5 │ 0.5  0.0
─────┼──────────
     │ 0.0  1.0

The Tableau type has the following fields

• name is a descriptive name of the tableau,
• o the order of the method,
• s the number of stages,
• a the coefficients,
• b the weights,
• c the nodes.

The following tableaus are implemented (prepend Tableau to the name to call the respective constructor):

• explicit: ExplicitEuler/ForwardEuler, ExplicitMidpoint, Heun2, Heun3, Ralston2, Ralston3, Runge2, Kutta3, RK21, RK22, RK31, RK32, RK4, RK41, RK42, RK416, RK438, RK5, SSPRK2, SSPRK3

• diagonally implicit: CrankNicolson, Crouzeix, KraaijevangerSpijker, QinZhang

• fully implicit: ImplicitEuler/BackwardEuler, ImplicitMidpoint, SRK3

In addition there exist functions to compute Gauss, Lobatto and Radau tableaus with an arbitrary number of stages s:

• TableauGauss(s)

• TableauLobattoIIIA(s), TableauLobattoIIIB(s), TableauLobattoIIIC(s), TableauLobattoIIIC̄(s), TableauLobattoIIID(s), TableauLobattoIIIE(s), TableauLobattoIIIF(s), TableauLobattoIIIG(s)

• TableauRadauIA(s), TableauRadauIIA(s)

All constructors take an optional type argument T, as in TableauExplicitMidpoint(T) or TableauGauss(T,s). The default type is Float64, but it can be set to other number types if needed, e.g., to Float32 for single precision or to the Dec128 type from DecFP.jl for quadruple precision. Internally, all tableaus are computed using BigFloat, providing high-accuracy coefficients as they are required for simulations in quadruple or higher precision. The internal precision can be set via setprecision(40), cf. the Julia Manual on Arbitrary Precision Arithmetic.

Custom Tableaus

If required, it is straight-forward to create a custom tableau. The tableau of Heun's method, for example, is defined as follows:

a = [[0.0  0.0]
     [1.0  0.0]]
b =  [0.5, 0.5]
c =  [0.0, 1.0]
o = 2

tab = Tableau(:heun, o, a, b, c)

Diagnostics

Currently, diagnostic functions for checking symmetry, symplecticity and the so-called simplifying assumptions are implemented:

• issymmetric(tab)
• issymplectic(tab)
• satisfies_simplifying_assumption_b(tab, σ=tab.s)
• satisfies_simplifying_assumption_c(tab, σ=tab.s)
• satisfies_simplifying_assumption_d(tab, σ=tab.s)

This list is expected to grow in the near future.

References

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

@misc{Kraus:2020:RungeKutta,
  title={RungeKutta.jl: Runge-Kutta Methods in Julia},
  author={Kraus, Michael},
  year={2020},
  howpublished={\url{https://github.com/JuliaGNI/RungeKutta.jl}},
  doi={10.5281/zenodo.4294923}
}