Support for purely parabolic PDEs via SemidiscretizationParabolic

[tristanmontoya]

The main purpose of this PR is to add SemidiscretizationParabolic to handle purely parabolic PDEs, including the new types LinearDiffusionEquation1D and LinearDiffusionEquation2D. It builds upon #2868, which renames "viscous" to "parabolic" when referring more generally to parabolic PDEs rather than fluid viscosity.

Implementation notes

• Uses only one equations struct and doesn't require an unused hyperbolic part
• rhs! is specialized on SemidiscretizationParabolic to call rhs_parabolic! only.
• Still uses cache and cache_parabolic analogously to SemidiscretizationHyperbolicParabolic
• We still require separate solver and solver_parabolic, with the latter used to dispatch the formulation for the parabolic terms, e.g., BR1 or LDG.
• Eventually, we may want to refactor SemidiscretizationHyperbolicParabolic to include modular hyperbolic and parabolic components, in addition to "base" equation-independent features. As a first step, however, I have not modified any of the structure for mixed hyperbolic-parabolic problems.

Elixirs changed/added

• examples/tree_1d_dgsem/elixir_diffusion_ldg.jl: migrated from zero-advection hyperbolic-parabolic setup to pure diffusion via LinearDiffusionEquation1D and SemidiscretizationParabolic.
• examples/tree_1d_dgsem/elixir_diffusion_ldg_newton_krylov.jl: updated the Newton-Krylov pure-diffusion example to use the dedicated parabolic API.
  examples/tree_2d_dgsem/elixir_diffusion_steady_state_linear_map.jl: switched the steady-state 2D diffusion example to LinearDiffusionEquation2D and SemidiscretizationParabolic.
• examples/tree_1d_dgsem/elixir_diffusion_ldg_amr_boundary_layer.jl: new pure-diffusion AMR example with a boundary layer and mixed Dirichlet/Neumann boundary conditions.

Benchmarks

For a simple 1D diffusion example, I get a speedup of just under 1.5x by using a purely parabolic semidiscretization rather than a hyperbolic-parabolic semidiscretization with zero advection velocity (which was previously the only way to solve parabolic problems). Using this benchmark script, I get the following output:

Environment info ===================================================
Julia threads: 1
BLAS threads: 4
Trixi.jl version: 0.16.1-DEV
OrdinaryDiffEqLowStorageRK version: 1.12.0
Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 10 × Apple M4
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)

Timing/consistency results =========================================
Parabolic solve median: 60.025 ms
Hyperbolic-parabolic solve median: 87.251 ms
Median ratio (hyperbolic-parabolic/parabolic): 1.454x
Final-state relative difference (hyperbolic-parabolic vs parabolic): 0.000e+00

Other changes

• I have added my name to the list of contributors in AUTHORS.md.
• The diffusion/heat equation is now listed in README.md as an example of a scalar conservation law, as it is the canonical parabolic problem and users should know that it is supported.
• NEWS.md has been updated to include this change as part of the v0.16 lifecycle.

LLM/AI usage

OpenAI Codex was used to generate some of the initial code. As the author of this PR, I have reviewed the code in its entirety and take full responsibility for its content.

[github-actions]

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Created with ❤️ by the Trixi.jl community.

[codecov]

Codecov Report

❌ Patch coverage is 97.91667% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 97.08%. Comparing base (16e465f) to head (d1f2b25).

Files with missing lines	Patch %	Lines
src/equations/linear_diffusion_equation_1d.jl	90.91%	1 Missing ⚠️
src/equations/linear_diffusion_equation_2d.jl	92.31%	1 Missing ⚠️
...semidiscretization/semidiscretization_parabolic.jl	98.65%	1 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff            @@
##             main    #2874    +/-   ##
========================================
  Coverage   97.08%   97.08%            
========================================
  Files         611      616     +5     
  Lines       47580    47690   +110     
========================================
+ Hits        46189    46297   +108     
- Misses       1391     1393     +2     

Flag	Coverage Δ
unittests	97.08% <97.92%> (+<0.01%)	⬆️

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[tristanmontoya]

Benchmarks now added to PR description.

[DanielDoehring]

LGTM!

[tristanmontoya]

@vchuravy I assume the buildkite failures are unrelated to this PR?
Edit: I guess it succeeded here eventually.

[vchuravy]

I assume the buildkite failures are unrelated to this PR?

CI for AMDGPU doesn't have very many runners so that can sometimes be a bit slower.

[ranocha]

Thanks!

[ranocha]

Do we really want this to be rhs! instead of rhs_parabolic!?

[tristanmontoya]

This way, we don't have to specialize semidiscretize, which assumes the function is called rhs! in the generic case. Otherwise, we'd have to make semidiscretize for SemidiscretizationParabolic do the exact same thing as for a generic AbstractSemidiscretization, except renaming rhs! to rhs_parabolic!.

I also think the naming makes intuitive sense in that it designates the "right-hand side for a parabolic semidiscretization" rather than "the parabolic right-hand side of a semidiscretization" (the latter for which rhs_parabolic! makes sense in the context of a hyperbolic-parabolic problem).

[ranocha]

For clarity, I would rather argue that our current rhs! should be rhs_hyperbolic!. This is how we basically use it. Using rhs_parabolic! for parabolic terms would appear more natural to me.
Maybe you can add this to the agenda of the next Trixi.jl meeting? I would like to get input from a few additional people.

[tristanmontoya]

Sure, I'll add it to the agenda. I see your point, and I'm not against the idea, but then we can't fall back to semidiscretize(::AbstractSemidiscretization) for both hyperbolic and parabolic problems unless we add another layer to make the outermost rhs! call dispatch directly to rhs_parabolic! or rhs_hyperbolic! as needed. Otherwise, I think there would be a lot of repeated code.

[ranocha]

Can't we have a small dispatch of the form default_rhs(::SemidiscretizationParabolic) = rhs_parabolic! and similarly for SemidiscretizationHyperbolic that is used in semidiscretize?

[ranocha]

Can you please synchronize

Trixi.jl/docs/src/index.md

Line 61 in df63d8c

* Several scalar conservation laws (e.g., linear advection, Burgers' equation, LWR traffic flow)

as well?