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BVP Solvers

The MIRK methods are recommended in most scenarios given their improved stability properties over the other methods. They have adaptivty and sparsity handling which allows for them to handle large-scale and difficult problems. However, they are not compatible with callbacks / event handling (i.e. discontinuities), and in such cases Shooting methods are required. There are single shooting methods and multiple shooting methods available in BoundaryValueDiffEq.jl. Shooting methods should be used with an appropriate ODE solver such as Shooting(Tsit5()) or MultipleShooting(5, FBDF()). Additionally, in many cases, single shooting method Shooting may be faster than collocation methods if it converges, though it is a lot less numerically robust. Multiple shooting method MultipleShooting is more stable and robust than single shooting method Shooting.

Full List of Methods

BoundaryValueDiffEq.jl

Shooting Methods

  • Shooting(odealg()) - A wrapper over initial value problem solvers, it reduces BVP to an initial value problem and solves the IVP.
  • MultipleShooting(N, odealg()) - A wrapper over initial value problem solvers, it reduces BVP to N initial value problems and solves these IVPs. Multiple Shooting usually maintains more numerical stability than Single Shooting.

MIRK Collocation Methods

All MIRK methods have defect control adaptivity by default which adapts the mesh (dt) automatically. This can be turned off via the keyword argument adaptive = false.

  • MIRK2 - A 2nd order collocation method using an implicit Runge-Kutta tableau with a sparse Jacobian.
  • MIRK3 - A 3rd order collocation method using an implicit Runge-Kutta tableau with a sparse Jacobian.
  • MIRK4 - A 4th order collocation method using an implicit Runge-Kutta tableau with a sparse Jacobian.
  • MIRK5 - A 5th order collocation method using an implicit Runge-Kutta tableau with a sparse Jacobian.
  • MIRK6 - A 6th order collocation method using an implicit Runge-Kutta tableau with a sparse Jacobian.

ODEInterface.jl

ODEInterface.jl can be used seamlessly with BoundaryValueDiffEq.jl, after we define our model using BVProblem or TwoPointBVProblem, we can directly call the solvers from ODEInterface.jl.

  • BVPM2 - FORTRAN code for solving two-point boundary value problems. BVPM2 is only compatible with TwoPointBVProblem.
  • BVPSOL - FORTRAN77 code which solves highly nonlinear two point boundary value problems using a local linear solver (condensing algorithm) or a global sparse linear solver for the solution of the arising linear subproblems, by Peter Deuflhard, Georg Bader, Lutz Weimann. BVPSOL should be used with TwoPointBVProblem and initial guess.