BoundaryValueDiffEqShooting

Single shooting method and multiple shooting method. To only use the Shooting methods form BoundaryValueDiffEq.jl, you need to install them use the Julia package manager:

using Pkg
Pkg.add("BoundaryValueDiffEqShooting")

Shooting algorithms operate on problem definitions and solver functions owned by SciMLBase, and require an ODE algorithm from its owning solver package:

using BoundaryValueDiffEqShooting: MultipleShooting, Shooting
using OrdinaryDiffEqTsit5: Tsit5
using SciMLBase: BVProblem, ReturnCode, solve

function f!(du, u, p, t)
    du[1] = u[2]
    du[2] = 0
end

function bc!(residual, u, p, t)
    residual[1] = u(0.0)[1] - 1
    residual[2] = u(1.0)[1]
end

prob = BVProblem(f!, bc!, [1.0, -1.0], (0.0, 1.0); nlls = Val(false))
sol = solve(prob, Shooting(Tsit5()); abstol = 1e-8)

@assert sol.retcode == ReturnCode.Success
@assert isapprox(sol(0.0)[1], 1.0; atol = 1e-6)
@assert isapprox(sol(1.0)[1], 0.0; atol = 1e-6)
# output

Full List of Methods

Shooting: Single shooting methods, reduces BVP to an initial value problem and solves the IVP.
MultipleShooting: Reduces BVP to an initial value problem and solves the IVP. Significantly more stable than Single Shooting.

Detailed Solvers Explanation

BoundaryValueDiffEqShooting.Shooting — Type

Shooting(ode_alg; kwargs...)
Shooting(ode_alg, nlsolve; kwargs...)
Shooting(; ode_alg = nothing, nlsolve = nothing, optimize = nothing, jac_alg = nothing) -> Shooting

Configures the single-shooting algorithm for a boundary value problem. Single shooting integrates one initial value problem and solves for the initial condition that satisfies the boundary conditions.

Arguments

ode_alg: algorithm used to solve the internal SciMLBase.ODEProblem. Pass this as the first positional argument or keyword argument. nothing selects a loaded polyalgorithm; otherwise an ODE algorithm must be supplied.
nlsolve: nonlinear-solver algorithm for the shooting residual. Its autodiff setting is superseded by jac_alg when a Jacobian algorithm is materialized.

Keywords

ode_alg = nothing: ODE algorithm, as described above.
nlsolve = nothing: nonlinear-solver algorithm, as described above.
optimize = nothing: optimization-solver algorithm used when the selected BVP solve path formulates the residual as an optimization problem.
jac_alg = nothing: BVPJacobianAlgorithm configuration. When omitted, the constructor derives it from nlsolve and the problem during solve initialization. For single shooting, only its diffmode setting is used; the default is AutoForwardDiff when applicable and otherwise AutoFiniteDiff.

Fields

ode_alg: configured ODE algorithm or nothing.
nlsolve: configured nonlinear-solver algorithm or nothing.
optimize: configured optimization-solver algorithm or nothing.
jac_alg::BVPJacobianAlgorithm: materialized Jacobian-algorithm configuration.

Returns

Shooting: an algorithm object accepted by SciMLBase.solve for a boundary value problem.

Examples

using BoundaryValueDiffEqShooting: Shooting
using OrdinaryDiffEqTsit5: Tsit5

alg = Shooting(Tsit5())
@assert alg isa Shooting
# output

source

BoundaryValueDiffEqShooting.MultipleShooting — Type

MultipleShooting(; nshoots::Int, ode_alg = nothing, nlsolve = nothing,
    optimize = nothing, grid_coarsening = true, jac_alg = nothing) -> MultipleShooting
MultipleShooting(nshoots::Int; kwargs...)
MultipleShooting(nshoots::Int, ode_alg; kwargs...)
MultipleShooting(nshoots::Int, ode_alg, nlsolve; kwargs...)

Configures the multiple-shooting algorithm for a boundary value problem. Multiple shooting integrates an IVP on nshoots subintervals and solves for their matching initial conditions; it is generally more stable than Shooting.