NSDERungeKutta.jl

An abstract type for parameters of embedded Runge-Kutta solvers.

An abstract type for Butcher tableaus.

An abstract type for parameters of Newton steps in implicit Runge-Kutta solvers.

An abstract type for caching intermediate computations in Runge-Kutta solvers.

An abstract type for Runge-Kutta solver parameters.

An abstract type for Runge-Kutta solutions of NSDEBase.AbstractInitialValueProblems.

An abstract type for Runge-Kutta solvers of NSDEBase.AbstractInitialValueProblems.

An abstract type for time-step sizes, useful for embedded solvers.

AdaptiveParameters <: AbstractAdaptiveParameters

A composite type for the parameters of an adaptive AbstractRungeKuttaSolver.

Constructors

AdaptiveParameters(εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100)

Arguments

• εₐ :: Real : absolute tolerance
• εᵣ :: Real : relative tolerance
• Mₙ :: Integer : maximum number of iterations

ButcherTableau <: AbstractButcherTableau

A composite type for the Butcher tableau of a Runge-Kutta solver:

\[\begin{array}{c|c} c & A \\ \hline p & b^\intercal \\ q & d^\intercal \end{array}\]

Constructors

ButcherTableau(A, b, c, s, p[, d, q])
ButcherTableau(tableau::AbstractMatrix{<:Real})

Arguments

• A :: AbstractMatrix{<:Real} : matrix of coefficients
• b :: AbstractVector{<:Real} : vector of weights
• c :: AbstractVector{<:Real} : vector of nodes
• s :: Integer : number of stages
• p :: Integer : order of accuracy
• d :: AbstractVector{<:Real} : embedding's vector of weights (can be nothing)
• q :: Integer : embedding's order of accuracy (can be nothing)
• b_dense :: AbstractMatrix{<:Real} : matrix where col j is powers of theta for stage j

Functions

butchertableau : return matrix of parameters.

DiagonallyImplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for diagonally-implicit solvers.

Constructors

DiagonallyImplicitRungeKuttaSolver(tableau, stepsize, newton[, adaptive])
DIRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• newton :: AbstractNewtonParameters
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::DiagonallyImplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::DiagonallyImplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

ExplicitExponentialRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for an exponential explicit AbstractRungeKuttaSolver.

Constructors

ExplicitExponentialRungeKuttaSolver(tableau, stepsize[, adaptive])
ExRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ExplicitExponentialRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ExplicitExponentialRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

ExplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for explicit solvers.

Constructors

ExplicitRungeKuttaSolver(tableau, stepsize[, adaptive])
ERK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ExplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ExplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

ImplicitExplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for implicit-explicit solvers.

Constructors

ImplicitExplicitRungeKuttaSolver(implicitableau, explicitableau, stepsize, newton[, adaptive])
IERK(args...; kwargs...)

Arguments

• implicitableau :: AbstractButcherTableau
• explicitableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• newton :: AbstractNewtonParameters
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ImplicitExplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ImplicitExplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

ImplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for implicit solvers.

Constructors

ImplicitRungeKuttaSolver(tableau, stepsize, newton[, adaptive])
IRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• newton :: AbstractNewtonParameters
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ImplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ImplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

NewtonParameters <: AbstractNewtonParameters

A composite type for the parameters of simplified Newton.

Constructors

NewtonParameters(; εᵣ=1e-3, Mₙ=10)

Arguments

• εᵣ :: Real : relative tolerance
• Mₙ :: Integer : maximum number of iterations

RungeKuttaSolution <: AbstractRungeKuttaSolution

A composite type for an AbstractRungeKuttaSolution obtained using an AbstractRungeKuttaSolver.

Constructors

RungeKuttaSolution(u, t, k)
RungeKuttaSolution(problem, solver; dense=false)

Arguments

• u :: AbstractVector{<:AbstractVector{<:Number}} : numerical solution
• t :: AbstractVector{<:Real} : time grid
• k :: AbstractVector{<:AbstractVector{<:AbstractVector{<:Number}}} : stages history (for dense output)

Functions

• extract : extract all values for a specific variable
• firstindex : get the first index
• getindex : get specified value(s) and time
• lastindex : get the last index
• length : get the number of time steps
• setindex! : set value(s) and time
• numtimesteps : get the number of time steps
• numvariables : get the number of variables

(solution::RungeKuttaSolution)(tₚ::Real, f::Function)

uses Hermite's cubic splines to interpolate solution and approximate its value at tₚ. Note that it needs the derivative function f(u, t), e.g. from an NSDEBase.AbstractRightHandSide subtype.

(solution::RungeKuttaSolution)(tₚ::Real, tableau::AbstractButcherTableau)

Evaluates the dense output solution at tₚ using the stored stages and tableau coefficients.

(solution::RungeKuttaSolution)(tₚ::Real)

interpolates solution using linear splines, approximating its value at tₚ.

StepSize <: AbstractStepSize

A composite type for the step-size a Runge-Kutta solver.

Constructors

StepSize(h::Real)

Functions

stepsize : returns (last) step-size

firstindex(solution::RungeKuttaSolution)

returns the first index of solution.

getindex(solution::RungeKuttaSolution, v::AbstractVector) :: RungeKuttaSolution

returns a new RungeKuttaSolution containing the fields of solution at the indices v.

getindex(solution::RungeKuttaSolution, i::Integer) :: RungeKuttaSolution

returns new a RungeKuttaSolution containing the fields of solution at index i.

lastindex(solution::RungeKuttaSolution)

returns the last index of solution.

length(solution::RungeKuttaSolution)

returns the number of time steps in solution.

setindex!(solution::RungeKuttaSolution, values::RungeKuttaSolution, v::AbstractVector)

stores the fields of values into the fields of solution at the specified indices v.

setindex!(solution::RungeKuttaSolution, values::RungeKuttaSolution, i::Integer)

stores the fields of values into the fields of solution at the specified index i. Assumes values contains data for a single time step (e.g. from getindex).

setindex!(solution::RungeKuttaSolution, values::Tuple, i::Integer)

stores the values from values into the fields of solution at the specified index i. If solution is dense (has k), values must be a 3-tuple (u, t, k). Otherwise, values must be a 2-tuple (u, t).

size(solution::RungeKuttaSolution)

returns a tuple containing the number of variables and time steps in solution.

solve(problem::AbstractInitialValueProblem, solver::AbstractRungeKuttaSolver; dense::Bool=false, kwargs...) :: RungeKuttaSolution

computes the solution of problem using solver.

BackwardEuler(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
ImplicitEuler(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 1st-order backward Euler method.

BogackiShampine(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 3rd-order Bogacki-Shampine method with 2nd-order error estimate.

Butcher5(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 5th-order Butcher method.

Butcher6(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 6th-order Butcher method.

Butcher7(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 7th-order Butcher method.

CrankNicolson(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
LobattoIIIA2(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order Crank-Nicolson method.

DiagonallyImplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for diagonally-implicit solvers.

Constructors

DiagonallyImplicitRungeKuttaSolver(tableau, stepsize, newton[, adaptive])
DIRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• newton :: AbstractNewtonParameters
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::DiagonallyImplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::DiagonallyImplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

DormandPrince54(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
DP54(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 5th-order Dormand-Prince method with 4th-order error estimate.

DormandPrince54(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
DP54(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 5th-order Dormand-Prince method with 4th-order error estimate.

ExplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for explicit solvers.

Constructors

ExplicitRungeKuttaSolver(tableau, stepsize[, adaptive])
ERK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ExplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ExplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

ETDEuler(; h::Real=0.0) :: ExplicitExponentialRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 1st-order stiff Exponential-Time-Differencing Euler method. ```

ETDRK4(; h::Real=0.0) :: ExplicitExponentialRungeKuttaSolver

returns an ExplicitExponentialRungeKuttaSolver for the 2nd-order stiff Exponential-Time-Differencing Runge-Kutta method.

Euler(; h::Real=0.0) :: ExplicitRungeKuttaSolver
ExplicitEuler(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 1st-order Euler method. ```

ExplicitExponentialRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for an exponential explicit AbstractRungeKuttaSolver.

Constructors

ExplicitExponentialRungeKuttaSolver(tableau, stepsize[, adaptive])
ExRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ExplicitExponentialRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ExplicitExponentialRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

Euler(; h::Real=0.0) :: ExplicitRungeKuttaSolver
ExplicitEuler(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 1st-order Euler method. ```

Midpoint(; h::Real=0.0) :: ExplicitRungeKuttaSolver
ExplicitMidpoint(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 2nd-order mid-point method.

Fehlberg45(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
F45(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 4th-order Fehlberg method with 5th-order error estimate.

Fehlberg78(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
F78(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 7th-order Fehlberg method with 8th-order error estimate.

Fehlberg45(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
F45(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 4th-order Fehlberg method with 5th-order error estimate.

Fehlberg78(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
F78(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 7th-order Fehlberg method with 8th-order error estimate.

ImplicitMidpoint(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
GaussLegendre2(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order implicit midpoint method.

GaussLegendre4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 4th-order Gauß-Legendre method.

GaussLegendre6(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 6th-order Gauß–Legendre method.

Heun2(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 2nd-order Heun method.

Heun3(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 3rd-order Heun method.

HeunEuler(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 2nd-order Heun-Euler method with 1st-order error estimate.

ExplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for explicit solvers.

Constructors

ExplicitRungeKuttaSolver(tableau, stepsize[, adaptive])
ERK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ExplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ExplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

IMEXEuler(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver
IMEXSSP1_111(args...; kwargs...) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 1st-order IMEXEuler method.

IMEXEuler(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver
IMEXSSP1_111(args...; kwargs...) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 1st-order IMEXEuler method.

IMEXSSP2_222(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 2nd-order IMEX-SSP2(2,2,2) L-stable scheme.

IMEXSSP2_322(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 2nd-order IMEX-SSP2(3,2,2) stiffly-accurate scheme.

IMEXSSP2_332(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 2nd-order IMEX-SSP2(3,3,2) stiffly-accurate scheme.

IMEXSSP3_332(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitExplicitRungeKuttaSolver

returns an ImplicitExplicitRungeKuttaSolver for the 3rd-order IMEX-SSP3(3,3,2) L-stable scheme.

ImplicitRungeKuttaSolver <: AbstractRungeKuttaSolver

A composite type for implicit solvers.

Constructors

ImplicitRungeKuttaSolver(tableau, stepsize, newton[, adaptive])
IRK(args...; kwargs...)

Arguments

• tableau :: AbstractButcherTableau
• stepsize :: AbstractStepSize
• newton :: AbstractNewtonParameters
• adaptive :: AbstractAdaptiveParameters

Methods

(solver::ImplicitRungeKuttaSolver)(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem) :: RungeKuttaSolution
(solver::ImplicitRungeKuttaSolver)(problem::AbstractInitialValueProblem) :: RungeKuttaSolution

returns the solution of a problem using solver.

BackwardEuler(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
ImplicitEuler(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 1st-order backward Euler method.

ImplicitMidpoint(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
GaussLegendre2(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order implicit midpoint method.

KuttaNystrom5(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 5th-order Kutta-Nyström method.

LobattoIII2(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order Lobatto III method.

LobattoIII4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 4th-order Lobatto III method.

CrankNicolson(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver
LobattoIIIA2(args...; kwargs...) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order Crank-Nicolson method.

LobattoIIIA4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 4th-order Lobatto IIIA method.

LobattoIIIB4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 4th-order Lobatto IIIB method.

LobattoIIIC2(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 2nd-order Lobatto IIIC method.

LobattoIIIC4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 4th-order Lobatto IIIC method.

Midpoint(; h::Real=0.0) :: ExplicitRungeKuttaSolver
ExplicitMidpoint(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 2nd-order mid-point method.

RungeKutta4(; h::Real=0.0) :: ExplicitRungeKuttaSolver
RK4(args...; kwargs...)

returns an ExplicitRungeKuttaSolver for the 4th-order Runge-Kutta method.

RadauI3(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 3rd-order Radau I method.

RadauI5(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 5th-order Radau I method.

RadauIA3(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 3rd-order Radau IA method.

RadauIA5(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 5th-order Radau IA method.

RadauII3(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 3rd-order Radau II method.

RadauII5(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 5th-order Radau II method.

RadauIIA3(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 3rd-order Radau IIA method.

RadauIIA5(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: ImplicitRungeKuttaSolver

returns an ImplicitRungeKuttaSolver for the 5th-order Radau IIA method.

Ralston2(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 2nd-order Ralston method.

Ralston3(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 3rd-order Ralston method.

Ralston4(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 4th-order Ralston method.

Rule38(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 4th-order 3/8-rule method.

RungeKutta3(; h::Real=0.0) :: ExplicitRungeKuttaSolver
RK3(args...; kwargs...)

returns an ExplicitRungeKuttaSolver for the 3rd-order Kutta method.

RungeKutta4(; h::Real=0.0) :: ExplicitRungeKuttaSolver
RK4(args...; kwargs...)

returns an ExplicitRungeKuttaSolver for the 4th-order Runge-Kutta method.

SDIRK2(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 2nd-order SDIRK method.

SDIRK3(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 3rd-order SDIRK method.

SDIRK4(; h::Real=0.0, εᵣ::Real=1e-3, Mₙ::Integer=10) :: DiagonallyImplicitRungeKuttaSolver

returns an DiagonallyImplicitRungeKuttaSolver for the 4th-order SDIRK method.

SSPRK3(; h::Real=0.0) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 3rd-order Strong-Stability-Preserving Runge-Kutta method.

Verner65(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
V65(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 6th-order Verner method with 5th-order error estimate.

Verner65(; h::Real=0.0, εₐ::Real=0.0, εᵣ::Real=1e-5, Mₙ::Integer=100, save_stepsizes::Bool=false) :: ExplicitRungeKuttaSolver
V65(args...; kwargs...) :: ExplicitRungeKuttaSolver

returns an ExplicitRungeKuttaSolver for the 6th-order Verner method with 5th-order error estimate.

butchertableau(solver::AbstractRungeKuttaSolver) :: AbstractMatrix

returns the Butcher tableau of a solver as a matrix.

compensated_sum(sum::T, addend::T, error::Ref{T}) where T<:AbstractFloat

Adds addend to sum using the Kahan-Babuška-Neumaier algorithm to minimize floating-point round-off error. The accumulated error is stored in the error Ref.

extract(solution::RungeKuttaSolution, v::AbstractVector) :: RungeKuttaSolution

returns the variables of solution indicated by the indices v.

extract(solution::RungeKuttaSolution, i::Integer) :: RungeKuttaSolution

returns the i-th variable of solution. i = 0 returns t.

extract(solution::RungeKuttaSolution) :: RungeKuttaSolution

returns all variables of solution, including t.

is_strictly_lower_triangular(A::AbstractMatrix) :: Bool

Returns true if all entries on and above the main diagonal of A are zero.

numtimesteps(solution::RungeKuttaSolution)

returns the number of time steps in solution.

numvariables(solution::RungeKuttaSolution)

returns the number of variables in solution.

solve!(solution::AbstractRungeKuttaSolution, problem::AbstractInitialValueProblem, solver::AbstractRungeKuttaSolver; dense::Bool=false) :: RungeKuttaSolution

computes the solution of problem using solver.

stability_function(Z::AbstractMatrix, tableau::AbstractButcherTableau) :: AbstractMatrix
stability_function(Z::AbstractMatrix, solver::AbstractRungeKuttaSolver) :: AbstractMatrix

Computes the matrix stability function $R(Z) = I + (b^\top \otimes Z) (I \otimes I - A \otimes Z)^{-1} (\mathbb{1} \otimes I)$. This builds and solves a Kronecker system of size $sN \times sN$; avoid for large systems.

stability_function(z::Number, tableau::AbstractButcherTableau) :: Number
stability_function(z::Number, solver::AbstractRungeKuttaSolver) :: Number

Computes the scalar stability function $R(z) = 1 + z b^\top (I - zA)^{-1} \mathbb{1}$.

stepsize(solver::AbstractRungeKuttaSolver) :: Real

returns the step-size of a solver.