Methods with Tabulated Coefficients
RungeKutta.jl provides tabulated coefficients for various explicit and implicit (both diagonally and fully implicit) Runge-Kutta methods. All constructors take an optional type argument, defaulting to Float64.
Explicit Methods
| Function and Aliases | Stages | Order |
|---|---|---|
TableauExplicitEuler, TableauForwardEuler | 1 | 1 |
TableauExplicitMidpoint | 2 | 2 |
TableauHeun2 | 2 | 2 |
TableauHeun3 | 3 | 3 |
TableauRalston2 | 2 | 2 |
TableauRalston3 | 3 | 3 |
TableauRunge, TableauRunge2 | 2 | 2 |
TableauKutta, TableauKutta3 | 3 | 3 |
TableauRK21 | 2 | 2 |
TableauRK22 | 2 | 2 |
TableauRK31 | 3 | 3 |
TableauRK32 | 3 | 3 |
TableauRK4 | 6 | 5 |
TableauRK41 | 4 | 4 |
TableauRK42 | 4 | 4 |
TableauRK416 | 4 | 4 |
TableauRK438 | 4 | 4 |
TableauRK5 | 6 | 5 |
TableauSSPRK3 | 3 | 3 |
Diagonally Implicit Methods
| Function and Aliases | Stages | Order |
|---|---|---|
TableauCrankNicolson | 2 | 2 |
TableauKraaijevangerSpijker | 2 | 2 |
TableauQinZhang | 2 | 2 |
TableauCrouzeix | 2 | 3 |
Fully Implicit Methods
| Function and Aliases | Stages | Order |
|---|---|---|
TableauImplicitEuler, TableauBackwardEuler | 1 | 1 |
TableauImplicitMidpoint | 2 | 2 |
TableauSRK3 | 3 | 4 |
Gauß, Lobatto and Radau Methods
The coefficients of the Gauß, Lobatto and Radau methods are computed on-the-fly by the following constructors:
| Function | Method | Order |
|---|---|---|
TableauGauss(s) | Gauß with s stages | $2s$ |
TableauLobattoIII(s) | Lobatto III with s stages | $2s-2$ |
TableauLobattoIIIA(s) | Lobatto IIIA with s stages | $2s-2$ |
TableauLobattoIIIB(s) | Lobatto IIIB with s stages | $2s-2$ |
TableauLobattoIIIC(s) | Lobatto IIIC with s stages | $2s-2$ |
TableauLobattoIIID(s) | Lobatto IIID with s stages | $2s-2$ |
TableauLobattoIIIE(s) | Lobatto IIIE with s stages | $2s-2$ |
TableauLobattoIIIF(s) | Lobatto IIIF with s stages | $2s$ |
TableauLobattoIIIG(s) | Lobatto IIIG with s stages | $2s$ |
TableauRadauIA(s) | Radau IA with s stages | $2s-1$ |
TableauRadauIB(s) | Radau IB with s stages | $2s-1$ |
TableauRadauIIA(s) | Radau IIA with s stages | $2s-1$ |
TableauRadauIIB(s) | Radau IIB with s stages | $2s-1$ |
The argument s refers to the number of stages ($s \ge 1$ for Gauß and $s \ge 2$ for all other methods). The type specifier can also be ommitted. For each method, a second constructor exists, where the first argument specifies the number type of the coefficients and the second argument the number of stages. Internally, all coefficients are computed using BigFloat and then converted to the requested number type, defaulting to Float64.