Extending Parareal

Start with a working serial solver and the built-in criteria. Extensions use qualified method names so that the package whose interface is extended is clear. The executable companion is examples/extensions.jl.

A custom maximum-seam criterion

using NSDERungeKutta, NSDETimeParallel, LinearAlgebra
struct MaxJump <: AbstractCriterion
    tolerance::Float64
end
function NSDETimeParallel.residual(c::MaxJump, cache, k)
    return maximum(norm(cache.U[n] - cache.F[n]) for n in eachindex(cache.U))
end
NSDETimeParallel.threshold(c::MaxJump) = c.tolerance
NSDETimeParallel.accept(c::MaxJump, value) = isfinite(value) && value <= c.tolerance
problem = Logistic(0.3, (0.0, 2.0))
p = Parareal(RK4(h=0.005), RK4(h=0.05); N=4, criterion=MaxJump(1e-8))
r = solve(problem, p)
@assert r.diagnostics.status in (:converged, :exact)

U[n] is start n; F[n] is the fine endpoint from chunk n-1, with F[1]=U[1]. Thus the first defect is zero. The loop also records the criterion threshold. Define threshold and accept explicitly, as here, or supply a field named ϵ and use their defaults. This criterion has the same unweighted maximum-defect target as WeightedDefect(p=Inf,weight=FixedWeight(1)); it illustrates the interface rather than adding a new numerical method.

Custom criteria work on bulk backends. A pipeline criterion needs a separate protocol; implementing residual does not make it pipeline-compatible. Cache fields used by an extension are implementation-facing and should be tested when upgrading the package.

Wrap an existing serial integrator

struct Wrapped{S} <: AbstractInitialValueSolver
    inner::S
end
NSDEBase.initialize_cache(problem::AbstractInitialValueProblem, s::Wrapped) =
    initialize_cache(problem, s.inner)
NSDEBase.initialize_solution(problem::AbstractInitialValueProblem, s::Wrapped; kw...) =
    initialize_solution(problem, s.inner; kw...)
NSDEBase.solve!(cache, sol, problem::AbstractInitialValueProblem, s::Wrapped) =
    solve!(cache, sol, problem, s.inner)
NSDEBase.solve(problem::AbstractInitialValueProblem, s::Wrapped; kw...) =
    solve(problem, s.inner; kw...)
NSDEBase.chunk_solver(s::Wrapped, duration::Real) = Wrapped(NSDEBase.chunk_solver(s.inner, duration))
pw = Parareal(Wrapped(RK4(h=0.007)), RK4(h=0.05); N=4, criterion=NoStop())
rw = solve(problem, pw)
@assert boundaryvalues(rw.lastiterate) == reference(problem, pw)

This wrapper delegates scheduling as well as stepping. A foreign integrator needs an adapter translating its cache, solution, cached solve!, and one-off solve operations, not just a new type name. Its solution must expose writable time/state storage and the indexing/interpolation operations the Parareal solution expects. Check early acceptance, non-dividing chunk lengths, exact completion, and failure behaviour. Scheduled endpoints should be reached directly; overshooting adapters may use interpolation and should not assume bitwise reference agreement automatically.

Throw a SolverFailure subtype for numerical failure. Leave programming errors as ordinary exceptions. Bulk non-MPI backends propagate ordinary errors; MPI logs ordinary chunk exceptions and returns :nonfinite to let peers complete communication. See Conventions for fatal errors and output invalidation.

For a custom problem, provide writable rhs, u0, tspan and an alias-preserving remake(problem; u0, tspan) as described in NSDEBase's guide. A custom solver for Stretch additionally needs the step-resizing interface when adaptation uses it.