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OrdinaryDiffEqFIRK

Fully Implicit Runge-Kutta (FIRK) methods for stiff differential equations requiring very high accuracy. These methods solve a fully coupled implicit system at each timestep, providing superior accuracy and stability compared to diagonally implicit methods.

Real Numbers Only

FIRK methods should only be used for problems defined on real numbers, not complex numbers.

Key Properties

FIRK methods provide:

  • Highest-order implicit methods (excluding extrapolation)
  • Superior accuracy for very low tolerance requirements (≤ 1e-9)
  • A-stable and L-stable behavior for stiff problems
  • Higher order per stage than SDIRK methods (order 2s+1 for s stages)
  • Special geometric properties (some methods are symplectic)
  • Excellent for small to medium systems with high accuracy requirements

When to Use FIRK Methods

These methods are recommended for:

  • Very low tolerance problems (1e-9 and below) where accuracy is paramount
  • Small to medium stiff systems (< 200 equations)
  • Problems requiring highest possible accuracy for implicit methods
  • Stiff problems where SDIRK order limitations (max order 5) are insufficient
  • Applications where computational cost is acceptable for maximum accuracy

Mathematical Background

RadauIIA methods are based on Gaussian collocation and achieve order 2s+1 for s stages, making them among the highest-order implicit methods available. They represent the ODE analog of Gaussian quadrature. For more details on recent advances in FIRK methods, see our paper: High-Order Adaptive Time Stepping for the Incompressible Navier-Stokes Equations.

Computational Considerations

Advantages

  • Higher accuracy per stage than diagonal methods
  • Better multithreading for small systems due to larger linear algebra operations
  • No order restrictions like SDIRK methods (which max out at order 5)

Disadvantages

  • Limited to real-valued problems - cannot be used for complex number systems
  • Higher implementation complexity compared to SDIRK methods

Solver Selection Guide

High accuracy requirements

  • AdaptiveRadau: Recommended - adaptive order method that automatically selects optimal order
  • RadauIIA5: 5th-order method, good balance of accuracy and efficiency
  • RadauIIA9: 9th-order method for extremely high accuracy requirements
  • RadauIIA3: 3rd-order method for moderate accuracy needs

System size considerations

  • Systems < 200: FIRK methods are competitive due to better multithreading
  • Systems > 200: Consider SDIRK or BDF methods instead

Performance Guidelines

  • Best for tolerances ≤ 1e-9 where high accuracy justifies the cost
  • Most efficient on small to medium systems where linear algebra cost is manageable
  • Should be tested against parallel implicit extrapolation methods which specialize in similar regimes
  • Compare with high-order SDIRK methods for borderline cases

Installation

To be able to access the solvers in OrdinaryDiffEqFIRK, you must first install them use the Julia package manager:

using Pkg
Pkg.add("OrdinaryDiffEqFIRK")

This will only install the solvers listed at the bottom of this page. If you want to explore other solvers for your problem, you will need to install some of the other libraries listed in the navigation bar on the left.

Example usage

using OrdinaryDiffEqFIRK

function lorenz!(du, u, p, t)
    du[1] = 10.0 * (u[2] - u[1])
    du[2] = u[1] * (28.0 - u[3]) - u[2]
    du[3] = u[1] * u[2] - (8 / 3) * u[3]
end
u0 = [1.0; 0.0; 0.0]
tspan = (0.0, 100.0)
prob = ODEProblem(lorenz!, u0, tspan)
sol = solve(prob, RadauIIA5())

Full list of solvers

OrdinaryDiffEqFIRK.RadauIIA3 — Type
RadauIIA3(; chunk_size = Val{0}(),
            autodiff = AutoForwardDiff(),
            standardtag = Val{true}(),
            concrete_jac = nothing,
            diff_type = Val{:forward},
            linsolve = nothing,
            precs = DEFAULT_PRECS,
            extrapolant = :dense,
            smooth_est = true,
            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.

Keyword Arguments

  • autodiff: Uses ADTypes.jl to specify whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to AutoForwardDiff() for automatic differentiation, which by default uses chunksize = 0, and thus uses the internal ForwardDiff.jl algorithm for the choice. To use FiniteDiff.jl, the AutoFiniteDiff() ADType can be used, which has a keyword argument fdtype with default value Val{:forward}(), and alternatives Val{:central}() and Val{:complex}().
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA3(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source
OrdinaryDiffEqFIRK.RadauIIA5 — Type
RadauIIA5(; chunk_size = Val{0}(),
            autodiff = AutoForwardDiff(),
            standardtag = Val{true}(),
            concrete_jac = nothing,
            diff_type = Val{:forward},
            linsolve = nothing,
            precs = DEFAULT_PRECS,
            extrapolant = :dense,
            smooth_est = true,
            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. 5th order method with excellent numerical stability. Good for highly stiff systems, problems requiring high-order implicit integration, systems with complex eigenvalue structures. Best for low tolerance stiff problems (<1e-9).

Keyword Arguments

  • autodiff: Uses ADTypes.jl to specify whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to AutoForwardDiff() for automatic differentiation, which by default uses chunksize = 0, and thus uses the internal ForwardDiff.jl algorithm for the choice. To use FiniteDiff.jl, the AutoFiniteDiff() ADType can be used, which has a keyword argument fdtype with default value Val{:forward}(), and alternatives Val{:central}() and Val{:complex}().
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA5(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source
OrdinaryDiffEqFIRK.RadauIIA9 — Type
RadauIIA9(; chunk_size = Val{0}(),
            autodiff = AutoForwardDiff(),
            standardtag = Val{true}(),
            concrete_jac = nothing,
            diff_type = Val{:forward},
            linsolve = nothing,
            precs = DEFAULT_PRECS,
            extrapolant = :dense,
            smooth_est = true,
            step_limiter! = trivial_limiter!)

Fully-Implicit Runge-Kutta Method. An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency. Similar to Hairer's SEULEX.

Keyword Arguments

  • autodiff: Uses ADTypes.jl to specify whether to use automatic differentiation via ForwardDiff.jl or finite differencing via FiniteDiff.jl. Defaults to AutoForwardDiff() for automatic differentiation, which by default uses chunksize = 0, and thus uses the internal ForwardDiff.jl algorithm for the choice. To use FiniteDiff.jl, the AutoFiniteDiff() ADType can be used, which has a keyword argument fdtype with default value Val{:forward}(), and alternatives Val{:central}() and Val{:complex}().
  • standardtag: Specifies whether to use package-specific tags instead of the ForwardDiff default function-specific tags. For more information, see this blog post. Defaults to Val{true}().
  • concrete_jac: Specifies whether a Jacobian should be constructed. Defaults to nothing, which means it will be chosen true/false depending on circumstances of the solver, such as whether a Krylov subspace method is used for linsolve.
  • linsolve: Any LinearSolve.jl compatible linear solver. For example, to use KLU.jl, specify RadauIIA9(linsolve = KLUFactorization()). When nothing is passed, uses DefaultLinearSolver.
  • precs: Any LinearSolve.jl-compatible preconditioner can be used as a left or right preconditioner. Preconditioners are specified by the Pl,Pr = precs(W,du,u,p,t,newW,Plprev,Prprev,solverdata) function where the arguments are defined as:
    • W: the current Jacobian of the nonlinear system. Specified as either $I - \gamma J$ or $I/\gamma - J$ depending on the algorithm. This will commonly be a WOperator type defined by OrdinaryDiffEq.jl. It is a lazy representation of the operator. Users can construct the W-matrix on demand by calling convert(AbstractMatrix,W) to receive an AbstractMatrix matching the jac_prototype.
    • du: the current ODE derivative
    • u: the current ODE state
    • p: the ODE parameters
    • t: the current ODE time
    • newW: a Bool which specifies whether the W matrix has been updated since the last call to precs. It is recommended that this is checked to only update the preconditioner when newW == true.
    • Plprev: the previous Pl.
    • Prprev: the previous Pr.
    • solverdata: Optional extra data the solvers can give to the precs function. Solver-dependent and subject to change.
    The return is a tuple (Pl,Pr) of the LinearSolve.jl-compatible preconditioners. To specify one-sided preconditioning, simply return nothing for the preconditioner which is not used. Additionally, precs must supply the dispatch:
    Pl, Pr = precs(W, du, u, p, t, ::Nothing, ::Nothing, ::Nothing, solverdata)
    which is used in the solver setup phase to construct the integrator type with the preconditioners (Pl,Pr). The default is precs=DEFAULT_PRECS where the default preconditioner function is defined as:
    DEFAULT_PRECS(W, du, u, p, t, newW, Plprev, Prprev, solverdata) = nothing, nothing
  • extrapolant: TBD
  • smooth_est: TBD
  • step_limiter!: function of the form limiter!(u, integrator, p, t)

References

@article{hairer1999stiff, title={Stiff differential equations solved by Radau methods}, author={Hairer, Ernst and Wanner, Gerhard}, journal={Journal of Computational and Applied Mathematics}, volume={111}, number={1-2}, pages={93–111}, year={1999}, publisher={Elsevier}}

source