Steady State Solvers

solve(prob::SteadyStateProblem, alg; kwargs)

Solves for the steady states in the problem defined by prob using the algorithm alg. If no algorithm is given, a default algorithm will be chosen.

Recommended Methods

Conversion to a NonlinearProblem is generally the fastest method. However, this will not guarantee the preferred root (the stable equilibrium), and thus if the preferred root is required, then it's recommended that one uses DynamicSS. For DynamicSS, often an adaptive stiff solver, like a Rosenbrock or BDF method (Rodas5 or QNDF), is a good way to allow for very large time steps as the steady state approaches.

The SteadyStateDiffEq.jl methods on a SteadyStateProblem respect the time definition in the nonlinear definition, i.e., u' = f(u, t) uses the correct values for t as the solution evolves. A conversion of a SteadyStateProblem to a NonlinearProblem replaces this with the nonlinear system u' = f(u, ∞), and thus the direct SteadyStateProblem approach can give different answers (i.e., the correct unique fixed point) on ODEs with non-autonomous dynamics.

If you have an unstable equilibrium and you want to solve for the unstable equilibrium, then DynamicSS will not converge to that equilibrium for any initial condition. However, Nonlinear Solvers don't suffer from this issue, and thus it's recommended to use a nonlinear solver if you want to solve for the unstable equilibrium.

Full List of Methods

Conversion to NonlinearProblem

Any SteadyStateProblem can be trivially converted to a NonlinearProblem via NonlinearProblem(prob::SteadyStateProblem). Using this approach, any of the solvers from the Nonlinear System Solvers page can be used. As a convenience, users can use:

• SSRootfind: A wrapper around NonlinearSolve.jl compliant solvers which converts the SteadyStateProblem to a NonlinearProblem and solves it.

SteadyStateDiffEq.jl

SteadyStateDiffEq.jl uses ODE solvers to iteratively approach the steady state. It is a very stable method for solving nonlinear systems, though often computationally more expensive than direct methods.

• DynamicSS : Uses an ODE solver to find the steady state. Automatically terminates when close to the steady state. DynamicSS(alg; tspan = Inf) requires that an ODE algorithm is given as the first argument. The absolute and relative tolerances specify the termination conditions on the derivative's closeness to zero. This internally uses the TerminateSteadyState callback from the Callback Library. The simulated time, for which the ODE is solved, can be limited by tspan. If tspan is a number, it is equivalent to passing (zero(tspan), tspan).

Example usage:

using NonlinearSolve, SteadyStateDiffEq, OrdinaryDiffEq
sol = solve(prob, DynamicSS(Tsit5()))

using Sundials
sol = solve(prob, DynamicSS(CVODE_BDF()), dt = 1.0)