Resolving Issue: Errors in Simple Handwritten PDEs as ODEs #1291

GodotMisogi · 2025-07-07T06:51:54Z

Checklist

Appropriate tests were added
Any code changes were done in a way that does not break public API
All documentation related to code changes were updated
The new code follows the
contributor guidelines, in particular the SciML Style Guide and
COLPRAC.
Any new documentation only uses public API

Additional context

This merges changes to resolve issue #1284 on the benchmarks of handwritten PDEs as ODEs via finite-difference and pseudo-spectral methods.

The package dependencies have been updated to remove unused ones, and the compat entries have been updated for Julia v1.10. The implementations have been tested on local machines (Macbook Pro M1 Pro on macOS Sequoia and a beefier CPU running Ubuntu 22.04 LTS) by running SciMLBenchmarks.weave_file on the relevant .jmd files.

At present, changes have been implemented for the following PDEs in one spatial and one time dimension:

Allen-Cahn equation
Burgers' equation
Kortewreg-de Vries equation
Kuramoto-Sivashinsky equation

The following changes have been implemented to the finite-difference method benchmarks:

The implementation errors in the PDEs have been corrected; some have been updated to reflect their particular characteristics.
Sparse matrices are constructed using SparseArrays.
The MatrixOperator interfaces are used from the updated SciMLOperators for SplitODEProblem.
Sketches of the equations have been added (but require upcoming refinements and updates).
Space-time heatmaps of the solutions instead of line plots.

These are functioning for certain solver algorithm choices (IMEXEuler, CNAB2, CLAF2, ETDRK2, SBDF2)

Commits will be progressively added in this pull (such as pseudo-spectral method updates), including revisions based on comments from the reviewers (@ChrisRackauckas).

ChrisRackauckas · 2025-07-07T10:16:48Z

For the psuedospectral, see the comment in #929. The ApproxFun dev mentioned we should change to OrthogonalPolynomials.jl for this part so we should take that recommendation.

GodotMisogi · 2025-07-12T06:41:50Z

@ChrisRackauckas Could you please run the workflow so I can check the logs for errors?

Allen-Cahn and Burgers' equations via Chebyshev pseudo-spectral methods have been implemented using ClassicalOrthogonalPolynomials.jl. These have been tested with some mass matrix solvers (Rodas3, Rodas4, Rodas5, GRK4T, GRK4A).
The Kuramoto-Sivashinsky and Korteweg-De Vries equations via Fourier pseudo-spectral methods are now working with ApproxFun.jl. These two cases will likely need to be reformulated to utilize Chebyshev interpolants with Neumann boundary conditions, allowing for the use of mass matrix solvers.

ChrisRackauckas · 2025-07-12T10:47:14Z

allowing for the use of mass matrix solvers

Why are mass matrices required here? Most of these should resolve to not have any algebraic constraints if the right spectral discretization is chosen. (Singular) Mass matrices (DAEs) would prohibit most of the solvers from being used. Though if that's a requirement for one of them, then so bit and that changes the solvers that should be chosen there (and would be an interesting contrast)

ChrisRackauckas · 2025-07-12T10:49:16Z

benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd

+L = 2.0  # Domain length
+xs, prob = allen_cahn(N, L)
+# @time sol = solve(prob, AutoVern7(RadauIIA5(autodiff=false)); dt=1e-2, abstol=1e-14, reltol=1e-14, adaptive=true)
+@time sol = solve(prob, Tsit5(); abstol=1e-14, reltol=1e-14, adaptive=true)


Explicit methods usually have issues with convergence due to order loss. I'm not sure changing to an explicit method for these for the reference will work out well. Though you'll see the in the work-precision diagram if you have a problem with the convergence, since all of the other methods will hit a wall and only converge to the error of the reference solution.

Good catch, I commented the wrong line in this commit. Thanks!

ChrisRackauckas · 2025-07-12T10:50:06Z

benchmarks/SimpleHandwrittenPDE/allen_cahn_fdm_wpd.jmd

+reltols = 0.1 .^ (1:3)
+multipliers = 0.5 .^ (0:3)
+setups = [
+    Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers),


I'd keep the same sets as above. A set showing all of the exponential integrators, all of the IMEX methods, all of the normal methods, and then a comparison of the best between the families.

Understood, planning to reincorporate the complete tests for the server builds as I can only prototype a few sets with the current local resources. Should these still be fixed time-steps or use adaptive time-stepping?

ChrisRackauckas · 2025-07-12T10:51:54Z

benchmarks/SimpleHandwrittenPDE/allen_cahn_spectral_wpd.jmd

+    Dict(:alg => Rodas4(), :dts => 1e-4 * multipliers),
+    Dict(:alg => Rodas5(), :dts => 1e-4 * multipliers),
+    Dict(:alg => GRK4T(), :dts => 1e-4 * multipliers),
+    Dict(:alg => GRK4A(), :dts => 1e-4 * multipliers),


the main ones to add to the mass matrices comparisons would be FBDF and QNDF. Also a DAEProblem formulation testing DASKR.jl and Sundials.IDA. You can benchmark multiple simultaneous problem implementations using the prob_choice setup, see the DAE benchmarks for how this is done.

Great to know; thanks for the tips!

…ors.jl

…artsOperators.jl

…racy

…r derivatives from SummationByPartsOperators.jl

GodotMisogi · 2025-07-18T23:21:09Z

@ChrisRackauckas Please run the workflow so I can check the logs for errors?

The problem descriptions, including equations, for all partial differential equation benchmark tests have been added.
Allen-Cahn and Burgers' equations via finite-difference methods have been implemented using SummationByPartsOperators.jl for conveniently specifying the accuracy of the derivative operator representations.
Allen-Cahn and Burgers' equations via Chebyshev pseudo-spectral methods have been implemented using ClassicalOrthogonalPolynomials.jl. The use of mass matrix solvers has been side-stepped by implementing LU factorization of the (here, non-singular) mass matrix and inverting the linear systems in the vector field implementations provided to SplitODEProblem.
The Kuramoto-Sivashinsky and Korteweg-De Vries equations via finite-difference and Fourier pseudo-spectral methods are implemented using SummationByPartsOperators.jl with periodic boundary conditions.
All tests have included the previous sets of benchmarks comparing implicit-explicit and exponential integrator methods, along with comparisons between the families. Some solvers have been excluded due to excessively long runtimes and some report instabilities in time-integration, which will need further diagnostics.

…others

ChrisRackauckas · 2025-07-20T15:43:26Z

benchmarks/SimpleHandwrittenPDE/kdv_fdm_wpd.jmd

-setups = [Dict(:alg => KenCarp5()),
-          Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense)),
-          Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())),
-          Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)),
-          Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers),
-          Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers)]
-labels = hcat("KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)",
-              "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)")
-@time wp = WorkPrecisionSet(prob,abstols,reltols,setups;
-                            print_names=true, names=labels,
-                            numruns=5, error_estimate=:l2,
-                            save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
-
-plot(wp, label=labels, markershape=:auto, title="Between family, medium order")
+abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
+reltols = 0.1 .^ (1:4)
+multipliers = 0.5 .^ (0:3)
+setups = [
+    Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
+    Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers),
+    Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers),


This is missing all of the good high order methods?

I should've specified that the high-tolerance solvers were tested. Currently testing the low-tolerance/high-order methods; will be included in the next few commits.

@ChrisRackauckas Please run the workflow on the latest commit? I think it's mostly ready.

…ciMLBenchmarks.jl into dev-pde-benchmarks

… FDM

ChrisRackauckas · 2025-08-01T16:08:23Z

Wow this is looking pretty good. Very thorough, everything looks correct. I think there's a few little tweaks to maybe do, like adding preconditioners on the Krylov methods, but what I'm going to do is merge it so it can build and it's easier to see it all together. I think it's like 95% or 99% done, and the last little bits I can probably do most of. But let's see what it looks like in the docs. This is a great change, using all of the latest PDE tools and everything!

GodotMisogi and others added 8 commits June 16, 2025 21:21

Initial fix tests

ade1fbf

FDM - SciMLOperators and plotting upgrades

6896952

Merge branch 'SciML:master' into dev-pde-benchmarks

81308f6

Burgers' FDM working

2e8a9bf

KdV FDM working

f63252c

KS FDM working

03c5b9b

Allen-Cahn FDM kinda working

f244a59

Updated dependencies

c71799a

GodotMisogi and others added 7 commits July 7, 2025 18:29

Merge branch 'SciML:master' into dev-pde-benchmarks

d2594ac

Burgers' spectral ApproxFun working

c7a6f3a

Burgers' spectral OCP working

83097f7

KdV pseudospectral ApproxFun kinda working

1a90bf4

KS pseudospectral ApproxFun working

2388581

Added DomainSets for scaling domains

33a4492

Allen-Cahn pseudospectral COP working

acc8166

ChrisRackauckas reviewed Jul 12, 2025

View reviewed changes

GodotMisogi added 9 commits July 18, 2025 01:49

Fixed difference operator accuracy issues with SummationByPartsOperat…

4f32412

…ors.jl

Used SummationByPartsOperators.jl and added more benchmarks

ef95ded

Added benchmarks to ClassicalOrthogonalPolynomials.jl cases

0d7eea1

Updated dependencies

e2b3051

Added problem description for Allen-Cahn FDM

aad85c6

Added problem description for Allen-Cahn pseudospectral

2b85e14

Added problem description for Burgers' FDM + swithced to SummationByP…

30e3f44

…artsOperators.jl

Added problem description for Burgers' pseudospectral

1998bb4

Added problem description for KdV FDM + changed derivative order accu…

6337711

…racy

GodotMisogi added 2 commits July 18, 2025 18:10

Added problem description for KdV and KS pseudospectral + used Fourie…

bad9540

…r derivatives from SummationByPartsOperators.jl

Added problem description for KS FDM + changed derivative order accuracy

127683d

GodotMisogi added 2 commits July 19, 2025 01:04

Reduced derivative order accuracy in Allen-Cahn FDM + minor edits in …

e17ad67

…others

Fixed typos

9884921

ChrisRackauckas reviewed Jul 20, 2025

View reviewed changes

GodotMisogi and others added 11 commits July 21, 2025 05:42

Added high-order benchmarks to Allen-Cahn and Burgers' tests

164ce6c

Tested higher-order methods; need to further diagnose KenCarp and ARKODE

8e03008

Fixed typo + tested KenCarp and ARKODE solvers in KS pseudo-spectral

90bfa8e

Some KenCarp tests working

0ffc04d

Updated coefficient for more interesting result

667a11b

KenCarp low tol kinda working on KdV pseudo-spectral

8392599

Reorganized high tolerance and low tolerance order in KdV pseudospectral

90a3e83

Removed DifferentialEquations.jl dependency

2056a26

Removed KenCarp for now in KdV FDM

5e1f0f6

Resolved Manifest.toml

27f2c28

Fixed typo in KS pseudo-spectral

be23be3

GodotMisogi closed this Aug 1, 2025

GodotMisogi force-pushed the dev-pde-benchmarks branch from be23be3 to 82b34f3 Compare August 1, 2025 07:22

Merge branch 'dev-pde-benchmarks' of https://github.com/GodotMisogi/S…

2267de9

…ciMLBenchmarks.jl into dev-pde-benchmarks

GodotMisogi reopened this Aug 1, 2025

GodotMisogi added 2 commits August 1, 2025 02:41

Updated Manifest.toml with merge

9d6a8e7

Reorganized high tolerance and low tolerance codeblocks in KS and KdV…

7e5cf3c

… FDM

ChrisRackauckas merged commit 0a5ccb8 into SciML:master Aug 1, 2025
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GodotMisogi mentioned this pull request Aug 1, 2025

Errors in Simple Handwritten PDEs as ODEs #1284

Closed

ChrisRackauckas mentioned this pull request Aug 2, 2025

Update Small Grants (Completion): "Simple Handwritten PDEs as ODEs" Benchmark Set" SciML/sciml.ai#171

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Resolving Issue: Errors in Simple Handwritten PDEs as ODEs #1291

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Resolving Issue: Errors in Simple Handwritten PDEs as ODEs #1291

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