QNDF gets stuck when solving for a steady state to high tolerances

[kockahonza]

Describe the bug 🐞

I am solving a SteadyState ODE problem with the DynamicSS solver using QNDF and the solver gets stuck when setting both abstol and reltol very small. The reason why I think this is a bug and not just the system taking a long time is that I suspect the algorithm gets stuck within a single QNDF step (as opposed to just taking many steps) meaning there is no way to interrupt it that I could find, and perhaps this seems like something that should not happen. I don't know how to debug this enough to give a precise reason for what I experience however the general observations are as follow:

• I've implemented a discrete callback that checks real time at each iteration and will exit after a set interval however this never triggers suggesting QNDF get stuck in the middle of one iteration.
• The solver cannot be interrupted, trying Control-C makes julia fail entirely - usually I just get back to the repl.
• QNDF works when abstol=reltol=1000*eps() and takes ~0.01s to solve the problem but changing them to 100*eps() makes it hang. I realize this might just be its limit but this seems like an extreme case, I have had this solver ran for over 2 days on a cluster.
• TRBDF2 works despite it usually being slower - at the 1000*eps() option it takes ~0.08s, but unlike QNDF it works even at 100*eps().
• In addition if I really push TRBDF2 beyond reason setting tolerances to eps() / 100, it still triggers my timer and the solver will exit after the specified time without a solution.

Expected behavior

I am aware this is a tricky subject but I suppose I would like something similar to how TRBDF2 works and make sure QNDF at least occasionally will make an iteration. On a separate note, in general having a real time limit for solvers would be great in my opinion as a solver getting stuck is sometimes unavoidable and hence having a way to recover from that would be great!

Minimal Reproducible Example 👇

This is fairly difficult as the problem is part of a growing codebase and relies on the parameters being a custom struct. However, I can point to a github with a notebook containing one, hopefully this should be easy to run! (Besides waiting 10min for the dependencies to install and precompile) THe github is: https://github.com/kockahonza/SSMC/tree/debug_qndf_differentialequations with the notebook containing a MWE being here: https://github.com/kockahonza/SSMC/blob/debug_qndf_differentialequations/notebooks/rg_unstructured.ipynb .

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

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  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
  [4c63d2b9] StatsFuns v1.5.0
  [3eaba693] StatsModels v0.7.4
  [9672c7b4] SteadyStateDiffEq v2.5.0
⌃ [789caeaf] StochasticDiffEq v6.79.0
  [7792a7ef] StrideArraysCore v0.5.7
  [892a3eda] StringManipulation v0.4.1
  [09ab397b] StructArrays v0.7.1
  [c3572dad] Sundials v4.28.0
  [2efcf032] SymbolicIndexingInterface v0.3.40
⌃ [19f23fe9] SymbolicLimits v0.2.1
⌅ [d1185830] SymbolicUtils v2.1.3
⌅ [0c5d862f] Symbolics v5.36.0
⌃ [7963cd39] SymbolicsMathLink v2.0.1
  [9ce81f87] TableMetadataTools v0.1.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.0
  [ed4db957] TaskLocalValues v0.1.2
  [62fd8b95] TensorCore v0.1.1
⌅ [8ea1fca8] TermInterface v0.4.1
  [5d786b92] TerminalLoggers v0.1.7
  [b189fb0b] ThreadPools v2.1.1
  [8290d209] ThreadingUtilities v0.5.4
  [731e570b] TiffImages v0.11.3
  [a759f4b9] TimerOutputs v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [981d1d27] TriplotBase v0.1.0
  [781d530d] TruncatedStacktraces v1.4.0
  [d265eb64] TypedSyntax v1.4.2
  [5c2747f8] URIs v1.5.2
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.22.1
  [a7c27f48] Unityper v0.1.6
  [19fa3120] VertexSafeGraphs v0.2.0
  [276b4fcb] WGLMakie v0.11.7
  [e3aaa7dc] WebP v0.1.3
  [eead4739] WidgetsBase v0.1.4
  [b8c1c048] WidthLimitedIO v1.0.1
  [efce3f68] WoodburyMatrices v1.0.0
  [6e34b625] Bzip2_jll v1.0.9+0
  [4e9b3aee] CRlibm_jll v1.0.1+0
  [83423d85] Cairo_jll v1.18.5+0
  [ee1fde0b] Dbus_jll v1.16.2+0
⌅ [04572ae6] Deno_jll v1.33.4+0
  [5ae413db] EarCut_jll v2.2.4+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v6.1.2+0
  [f5851436] FFTW_jll v3.3.11+0
  [a3f928ae] Fontconfig_jll v2.16.0+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [78b55507] Gettext_jll v0.21.0+0
  [59f7168a] Giflib_jll v5.2.3+0
  [7746bdde] Glib_jll v2.84.0+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [3c863552] Graphviz_jll v2.50.0+1
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [905a6f67] Imath_jll v3.1.11+0
  [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.2+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [aae0fff6] LSODA_jll v0.1.2+0
  [dd4b983a] LZO_jll v2.10.3+0
  [81d17ec3] L_BFGS_B_jll v3.0.1+0
  [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.0+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.41.0+0
  [856f044c] MKL_jll v2025.0.1+1
  [e7412a2a] Ogg_jll v1.3.5+1
  [6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.29+0
  [18a262bb] OpenEXR_jll v3.2.4+0
  [458c3c95] OpenSSL_jll v3.5.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.3.3+0
  [36c8627f] Pango_jll v1.56.3+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a2964d1f] Wayland_jll v1.23.1+0
  [2381bf8a] Wayland_protocols_jll v1.36.0+0
⌅ [02c8fc9c] XML2_jll v2.13.6+1
  [ffd25f8a] XZ_jll v5.8.1+0
  [4f6342f7] Xorg_libX11_jll v1.8.12+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.7+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.1+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.6+0
  [ec84b674] Xorg_libXrandr_jll v1.5.5+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.3+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [9a68df92] isoband_jll v0.2.3+0
  [a4ae2306] libaom_jll v3.11.0+0
  [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [b53b4c65] libpng_jll v1.6.48+0
  [075b6546] libsixel_jll v1.10.5+0
  [f27f6e37] libvorbis_jll v1.3.7+2
  [c5f90fcd] libwebp_jll v1.5.0+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
  [1270edf5] x264_jll v10164.0.1+0
⌅ [dfaa095f] x265_jll v3.6.0+0
  [d8fb68d0] xkbcommon_jll v1.8.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [8bf52ea8] CRC32c
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+4
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2

• Output of versioninfo()

Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 22 × Intel(R) Core(TM) Ultra 9 185H
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, goldmont)
Threads: 20 default, 0 interactive, 10 GC (on 22 virtual cores)

Additional context

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