BVP based on orthogonal collocation

• BVP based on orthogonal collocation
• • Mathematical formulation
  • Usage
  • Mesh adaptation
  • Jacobians
  • References

The Collocation method discretizes the BVP using orthogonal collocation on Ntst mesh intervals with polynomials of degree m. This is the most precise discretization method and supports automatic mesh adaptation during continuation.

Mathematical formulation

We look for solutions of

\[u'(t) = F(u(t), p),\quad u\in\mathbb R^n,\ t\in[t_0,t_f]\]

with boundary conditions $g(u(t_0), u(t_f), p)=0$.

The time domain is partitioned into $N_{tst}$ intervals. On each interval, the solution is approximated by a polynomial of degree $m$ and the residual is enforced at $m$ Gauss-Legendre collocation points.

Usage

using BifurcationKit
const BK = BifurcationKit

F(u, p) = [u[2], -p.ω^2 * u[1]]
g(u0, u1, p) = [u0[1], u1[1]]

model = BK.BVP.BVPModel(F, g; n=2)
disc = BK.BVP.Collocation(Ntst=20, m=4)
bvp = BK.BVP.discretize(model, disc)

x0 = BK.BVP.generate_solution(bvp, t -> [cos(t), sin(t)])
prob = BK.BVP.BVPBifProblem(bvp, x0, (ω=1.0,), (@optic _.ω))

br = continuation(prob, PALC(), ContinuationPar())

Mesh adaptation

The mesh can be automatically adapted during continuation to better resolve the solution:

disc = BK.BVP.Collocation(Ntst=20, m=4, meshadapt=true,
    K=100.0, update_every_step=1)

The adaptation works by equidistributing the collocation error across mesh intervals. It is triggered every update_every_step steps when the Newton solver has converged and the continuation step is not inside a bisection.

Jacobians

Two analytical Jacobian types are available:

• DenseAnalytical(): dense analytical Jacobian, suitable for small to medium systems
• FullSparse(): sparse analytical Jacobian, suitable for large sparse systems

prob_ana = BK.BVP.BVPBifProblem(bvp, x0, params, (@optic _.a);
    jacobian = BK.DenseAnalytical())

References