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BoundaryValueDiffEqMIRK

Monotonic Implicit Runge Kutta(MIRK) Methods. To only use the MIRK methods form BoundaryValueDiffEq.jl, you need to install them use the Julia package manager:

using Pkg
Pkg.add("BoundaryValueDiffEqFIRK")
solve(prob::BVProblem, alg, dt; kwargs...)
solve(prob::TwoPointBVProblem, alg, dt; kwargs...)

Full List of Methods

  • MIRK2: 2 stage Monotonic Implicit Runge-Kutta method, with defect control adaptivity.
  • MIRK3: 3 stage Monotonic Implicit Runge-Kutta method, with defect control adaptivity.
  • MIRK4: 4 stage Monotonic Implicit Runge-Kutta method, with defect control adaptivity.
  • MIRK5: 5 stage Monotonic Implicit Runge-Kutta method, with defect control adaptivity.
  • MIRK6: 6 stage Monotonic Implicit Runge-Kutta method, with defect control adaptivity.

Detailed Solvers Explanation

BoundaryValueDiffEqMIRK.MIRK2 — Type
MIRK2(; nlsolve = NewtonRaphson(), jac_alg = BVPJacobianAlgorithm(),
        defect_threshold = 0.1, max_num_subintervals = 3000)

2th order Monotonic Implicit Runge Kutta method.

Keyword Arguments

  • nlsolve: Internal Nonlinear solver. Any solver which conforms to the SciML NonlinearProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used.

  • optimize: Internal Optimization solver. Any solver which conforms to the SciML OptimizationProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used. Optimization solvers should first be loaded to allow this functionality.

  • jac_alg: Jacobian Algorithm used for the nonlinear solver. Defaults to BVPJacobianAlgorithm(), which automatically decides the best algorithm to use based on the input types and problem type.

    • For TwoPointBVProblem, only diffmode is used (defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff())).
    • For BVProblem, bc_diffmode and nonbc_diffmode are used. For nonbc_diffmode defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff()). For bc_diffmode, defaults to AutoForwardDiff if possible else AutoFiniteDiff.

  • defect_threshold: Threshold for defect control.

  • max_num_subintervals: Number of maximal subintervals, default as 3000.

Note

For type-stability, the chunksizes for ForwardDiff ADTypes in BVPJacobianAlgorithm must be provided.

References

@article{Enright1996RungeKuttaSW,
    title={Runge-Kutta Software with Defect Control for Boundary Value ODEs},
    author={Wayne H. Enright and Paul H. Muir},
    journal={SIAM J. Sci. Comput.},
    year={1996},
    volume={17},
    pages={479-497}
}
source
BoundaryValueDiffEqMIRK.MIRK3 — Type
MIRK3(; nlsolve = NewtonRaphson(), jac_alg = BVPJacobianAlgorithm(),
        defect_threshold = 0.1, max_num_subintervals = 3000)

3th order Monotonic Implicit Runge Kutta method.

Keyword Arguments

  • nlsolve: Internal Nonlinear solver. Any solver which conforms to the SciML NonlinearProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used.

  • optimize: Internal Optimization solver. Any solver which conforms to the SciML OptimizationProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used. Optimization solvers should first be loaded to allow this functionality.

  • jac_alg: Jacobian Algorithm used for the nonlinear solver. Defaults to BVPJacobianAlgorithm(), which automatically decides the best algorithm to use based on the input types and problem type.

    • For TwoPointBVProblem, only diffmode is used (defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff())).
    • For BVProblem, bc_diffmode and nonbc_diffmode are used. For nonbc_diffmode defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff()). For bc_diffmode, defaults to AutoForwardDiff if possible else AutoFiniteDiff.

  • defect_threshold: Threshold for defect control.

  • max_num_subintervals: Number of maximal subintervals, default as 3000.

Note

For type-stability, the chunksizes for ForwardDiff ADTypes in BVPJacobianAlgorithm must be provided.

References

@article{Enright1996RungeKuttaSW,
    title={Runge-Kutta Software with Defect Control for Boundary Value ODEs},
    author={Wayne H. Enright and Paul H. Muir},
    journal={SIAM J. Sci. Comput.},
    year={1996},
    volume={17},
    pages={479-497}
}
source
BoundaryValueDiffEqMIRK.MIRK4 — Type
MIRK4(; nlsolve = NewtonRaphson(), jac_alg = BVPJacobianAlgorithm(),
        defect_threshold = 0.1, max_num_subintervals = 3000)

4th order Monotonic Implicit Runge Kutta method.

Keyword Arguments

  • nlsolve: Internal Nonlinear solver. Any solver which conforms to the SciML NonlinearProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used.

  • optimize: Internal Optimization solver. Any solver which conforms to the SciML OptimizationProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used. Optimization solvers should first be loaded to allow this functionality.

  • jac_alg: Jacobian Algorithm used for the nonlinear solver. Defaults to BVPJacobianAlgorithm(), which automatically decides the best algorithm to use based on the input types and problem type.

    • For TwoPointBVProblem, only diffmode is used (defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff())).
    • For BVProblem, bc_diffmode and nonbc_diffmode are used. For nonbc_diffmode defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff()). For bc_diffmode, defaults to AutoForwardDiff if possible else AutoFiniteDiff.

  • defect_threshold: Threshold for defect control.

  • max_num_subintervals: Number of maximal subintervals, default as 3000.

Note

For type-stability, the chunksizes for ForwardDiff ADTypes in BVPJacobianAlgorithm must be provided.

References

@article{Enright1996RungeKuttaSW,
    title={Runge-Kutta Software with Defect Control for Boundary Value ODEs},
    author={Wayne H. Enright and Paul H. Muir},
    journal={SIAM J. Sci. Comput.},
    year={1996},
    volume={17},
    pages={479-497}
}
source
BoundaryValueDiffEqMIRK.MIRK5 — Type
MIRK5(; nlsolve = NewtonRaphson(), jac_alg = BVPJacobianAlgorithm(),
        defect_threshold = 0.1, max_num_subintervals = 3000)

5th order Monotonic Implicit Runge Kutta method.

Keyword Arguments

  • nlsolve: Internal Nonlinear solver. Any solver which conforms to the SciML NonlinearProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used.

  • optimize: Internal Optimization solver. Any solver which conforms to the SciML OptimizationProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used. Optimization solvers should first be loaded to allow this functionality.

  • jac_alg: Jacobian Algorithm used for the nonlinear solver. Defaults to BVPJacobianAlgorithm(), which automatically decides the best algorithm to use based on the input types and problem type.

    • For TwoPointBVProblem, only diffmode is used (defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff())).
    • For BVProblem, bc_diffmode and nonbc_diffmode are used. For nonbc_diffmode defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff()). For bc_diffmode, defaults to AutoForwardDiff if possible else AutoFiniteDiff.

  • defect_threshold: Threshold for defect control.

  • max_num_subintervals: Number of maximal subintervals, default as 3000.

Note

For type-stability, the chunksizes for ForwardDiff ADTypes in BVPJacobianAlgorithm must be provided.

References

@article{Enright1996RungeKuttaSW,
    title={Runge-Kutta Software with Defect Control for Boundary Value ODEs},
    author={Wayne H. Enright and Paul H. Muir},
    journal={SIAM J. Sci. Comput.},
    year={1996},
    volume={17},
    pages={479-497}
}
source
BoundaryValueDiffEqMIRK.MIRK6 — Type
MIRK6(; nlsolve = NewtonRaphson(), jac_alg = BVPJacobianAlgorithm(),
        defect_threshold = 0.1, max_num_subintervals = 3000)

6th order Monotonic Implicit Runge Kutta method.

Keyword Arguments

  • nlsolve: Internal Nonlinear solver. Any solver which conforms to the SciML NonlinearProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used.

  • optimize: Internal Optimization solver. Any solver which conforms to the SciML OptimizationProblem interface can be used. Note that any autodiff argument for the solver will be ignored and a custom jacobian algorithm will be used. Optimization solvers should first be loaded to allow this functionality.

  • jac_alg: Jacobian Algorithm used for the nonlinear solver. Defaults to BVPJacobianAlgorithm(), which automatically decides the best algorithm to use based on the input types and problem type.

    • For TwoPointBVProblem, only diffmode is used (defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff())).
    • For BVProblem, bc_diffmode and nonbc_diffmode are used. For nonbc_diffmode defaults to AutoSparse(AutoForwardDiff()) if possible else AutoSparse(AutoFiniteDiff()). For bc_diffmode, defaults to AutoForwardDiff if possible else AutoFiniteDiff.

  • defect_threshold: Threshold for defect control.

  • max_num_subintervals: Number of maximal subintervals, default as 3000.

Note

For type-stability, the chunksizes for ForwardDiff ADTypes in BVPJacobianAlgorithm must be provided.

References

@article{Enright1996RungeKuttaSW,
    title={Runge-Kutta Software with Defect Control for Boundary Value ODEs},
    author={Wayne H. Enright and Paul H. Muir},
    journal={SIAM J. Sci. Comput.},
    year={1996},
    volume={17},
    pages={479-497}
}
source