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BVP Solvers

solve(prob::BVProblem, alg, dt; kwargs)
solve(prob::TwoPointBVProblem, alg, dt; kwargs)
solve(prob::SecondOrderBVProblem, alg, dt; kwargs)
solve(prob::SecondOrderTwoPointBVProblem, alg, dt; kwargs)

Solves the BVP defined by prob using the algorithm alg. All algorithms except Shooting and MultipleShooting methods should specify a dt which is the step size for the discretized mesh.

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(Monotonic Implicit Runge-Kutta) 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.

FIRK(Fully Implicit Runge-Kutta) methods

Similar to MIRK methods, fully implicit Runge-Kutta methods construct nonlinear problems from the collocation equations of a BVP and solve such nonlinear systems to obtain numerical solutions of BVP. When solving large boundary value problems, choose a nested NonlinearSolve.jl solver by setting nested_nlsolve=true in FIRK solvers can achieve better performance.

  • LobattoIIIa2 - A 2nd stage LobattoIIIa collocation method.

  • LobattoIIIa3 - A 3rd stage LobattoIIIa collocation method.

  • LobattoIIIa4 - A 4th stage LobattoIIIa collocation method.

  • LobattoIIIa5 - A 5th stage LobattoIIIa collocation method.

  • LobattoIIIb2 - A 2nd stage LobattoIIIa collocation method, doesn't support defect control adaptivity.

  • LobattoIIIb3 - A 3rd stage LobattoIIIa collocation method.

  • LobattoIIIb4 - A 4th stage LobattoIIIa collocation method.

  • LobattoIIIb5 - A 5th stage LobattoIIIa collocation method.

  • LobattoIIIc2 - A 2nd stage LobattoIIIa collocation method, doesn't support defect control adaptivity.

  • LobattoIIIc3 - A 3rd stage LobattoIIIa collocation method.

  • LobattoIIIc4 - A 4th stage LobattoIIIa collocation method.

  • LobattoIIIc5 - A 5th stage LobattoIIIa collocation method.

  • RadauIIa1 - A 1st stage Radau collocation method, doesn't support defect control adaptivity.

  • RadauIIa2 - A 2nd stage Radau collocation method.

  • RadauIIa3 - A 3rd stage Radau collocation method.

  • RadauIIa5 - A 5th stage Radau collocation method.

  • RadauIIa7 - A 7th stage Radau collocation method.

Gauss Legendre collocation methods

The Ascher collocation methods are similar with MIRK and FIRK methods but have extension for BVDAE prblem solving, the error control is based on instead of defect control adaptivity.

  • Ascher1 - A 1st stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher2 - A 2nd stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher3 - A 3rd stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher4 - A 4th stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher5 - A 5th stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher6 - A 6th stage Gauss Legendre collocation method with Ascher's error control adaptivity.
  • Ascher7 - A 7th stage Gauss Legendre collocation method with Ascher's error control adaptivity.

MIRKN(Monotonic Implicit Runge-Kutta-Nystöm) methods

  • MIRKN4 - A 4th order collocation method using an implicit Runge-Kutta-Nyström tableau without defect control adaptivity.
  • MIRKN6 - A 6th order collocation method using an implicit Runge-Kutta-Nyström tableau without defect control adaptivity.

SimpleBoundaryValueDiffEq.jl

  • SimpleMIRK4 - A simplified 4th order collocation method using an implicit Runge-Kutta tableau.
  • SimpleMIRK5 - A simplified 5th order collocation method using an implicit Runge-Kutta tableau.
  • SimpleMIRK6 - A simplified 6th order collocation method using an implicit Runge-Kutta tableau.
  • SimpleShooting - A simplified single Shooting method.

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.
  • COLNEW - A Fortran77 code solves a multi-points boundary value problems for a mixed order system of ODEs by Uri Ascher and Georg Bader. It incorporates a new basis representation replacing b-splines, and improvements for the linear and nonlinear algebraic equation solvers. COLNEW support TwoPointBVProblem by default. To solve multi-points BVP using COLNEW, special form of multi-points boundary conditions should be provided by COLNEW(bc_func, dbc_func, zeta) where bc_func(i, z, res) is the multi-points boundary conditions, dbc_func(i, z, dbc) is the i-th row of jacobian of boundary conditions.