Tutorial: Solution of Robertson problem

In this tutorial we show that MPRK schemes can be used to integrate stiff problems. We also show how callbacks can be used to change the time step in non-adaptive schemes.

Definition of the production-destruction system

The well known Robertson problem is given by

\[\begin{aligned} u_1' &= -0.04u_1+10^4 u_2u_3, & u_1(0)&=1,\\ u_2' &= 0.04u_1-10^4 u_2u_3-3⋅10^7 u_2^2, & u_2(0)&=0 \\ u_3' &= 3⋅10^7 u_2^2, & u_3(0)&=0. \end{aligned}\]

The time domain of interest is $t\in[0,10^{11}]$, because of which some kind of adaptive time stepping is required.

The model can be represented as a conservative PDS with production terms

\[\begin{aligned} p_{12}(t,\mathbf{u}) &= 10^4u_2u_3,& p_{21}(t,\mathbf{u}) &= 0.04u_1, & p_{32}(t,\mathbf{u}) &= 3⋅10^7u_2^2, \end{aligned}\]

whereby production terms not listed have the value zero. Since the PDS is conservative, we have $d_{ij}=p_{ji}$ and the system is fully determined by the production matrix $\mathbf P=(p_{ij})$.

Solution of the production-destruction system

Now we are ready to define a ConservativePDSProblem and to solve this problem with any method of PositiveIntegrators.jl or OrdinaryDiffEq.jl which is suited for stiff problems.

Since this PDS consists of only three differential equations we provide an out-of-place implementation for the production matrix. Furthermore, we use static arrays from StaticArrays.jl for additional efficiency. See also the tutorials on the solution of an NPZD model or an stratospheric reaction problem.

using PositiveIntegrators, StaticArrays

function prod(u, p, t)
    @SMatrix [0.0 1e4*u[2]*u[3] 0.0;
              4e-2*u[1] 0.0 0.0;
              0.0 3e7*u[2]^2 0.0]
end
u0 = @SVector [1.0, 0.0, 0.0]
tspan = (0.0, 1.0e11)
prob = ConservativePDSProblem(prod, u0, tspan)

sol = solve(prob, MPRK43I(1.0, 0.5))

using Plots

plot(sol, tspan = (1e-6, 1e11),  xaxis = :log,
     idxs = [(0, 1), ((x, y) -> (x, 1e4 .* y), 0, 2), (0, 3)],
     label = ["u₁" "10⁴u₂" "u₃"])

PositiveIntegrators.jl provides the function isnonnegative (and also isnegative) to check if the solution is actually nonnegative, as expected from an MPRK scheme.

isnonnegative(sol)

true

Using callbacks to solve the Robertson problem with non-adatpive schemes

The SSPMPRK43() scheme is only available with fixed time stepping. With a scheme like this, it would take a huge amount of time to solve the Robertson problem, since the time step must be chosen very small to accurately solve the problem in its initial phase. However, the use of a callback allows us to modify the time step size after each step, which makes a solution with a fixed step method possible.

In the following example the callback increases the time step size by a factor of 1.5 after each time step.

using OrdinaryDiffEq

stepsize_callback = DiscreteCallback(
    Returns(true), # adapt the step size after every time step
    integrator -> set_proposed_dt!(integrator, 1.5 * get_proposed_dt(integrator));
    save_positions = (false, false),
    initialize = (c, u, t, integrator) -> set_proposed_dt!(integrator, 1.0e-5))
sol_cb = solve(prob, SSPMPRK43(); dt = Inf, callback =  stepsize_callback);

plot(sol_cb, tspan = (1e-6, 1e11),  xaxis = :log,
     idxs = [(0, 1), ((x, y) -> (x, 1e4 .* y), 0, 2), (0, 3)],
     label = ["u₁" "10⁴u₂" "u₃"])

This solution is also nonnegative.

isnonnegative(sol_cb)

true

Package versions

These results were obtained using the following versions.

using InteractiveUtils
versioninfo()
println()

using Pkg
Pkg.status(["PositiveIntegrators", "StaticArrays", "LinearSolve", "OrdinaryDiffEq"],
           mode=PKGMODE_MANIFEST)

Julia Version 1.11.3
Commit d63adeda50d (2025-01-21 19:42 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager

Status `~/work/PositiveIntegrators.jl/PositiveIntegrators.jl/docs/Manifest.toml`
  [7ed4a6bd] LinearSolve v2.38.0
  [1dea7af3] OrdinaryDiffEq v6.90.1
  [d1b20bf0] PositiveIntegrators v0.2.6 `~/work/PositiveIntegrators.jl/PositiveIntegrators.jl`
  [90137ffa] StaticArrays v1.9.11