GSoC 2026 · MethodOfLines.jl · Final Work Product
August 2026 · Official submission
MethodOfLines.jl discretizes symbolic PDE systems into ODE problems using finite differences. When I started, the high-order WENO advection scheme only worked on uniform grids, and the non-uniform code paths in general had little test coverage. My goal for the summer was to bring native non-uniform grid support to the WENO-5 scheme — from the core stencil math to boundaries, interfaces, tests, benchmarks, and documentation — without changing anything for existing uniform-grid users.
Roughly in order:
Started with tests. Added a regression suite for the existing upwind scheme on non-uniform grids (#562) so I had a safe baseline before touching any scheme code.
Set up the dispatch architecture (#542): uniform grids keep taking the existing path, vector grids route to the new one.
Wrote the non-uniform WENO-5 core (#581): stencil weights and smoothness indicators computed from the actual grid coordinates instead of assuming constant spacing. The kernel is zero-allocation and checked against exact solutions.
Added boundary handling (#582) with one-sided stencils near the domain edges, and wired everything into the discretizer (#583) so it works through the normal discretize workflow.
Added interface and periodic boundary support, both for WENO (#602) and for the upwind scheme (#572). This includes joining two subdomains that use independently generated non-uniform grids.
Validated and measured it: an MMS convergence test suite (#591) and a benchmark suite that now runs on every PR (#609).
Documented it: a tutorial (#615) and a showcase page with two worked examples against exact solutions (#616), also published as a blog post.
Along the way I helped keep CI green with some smaller fixes that weren't strictly part of the project but were blocking reviews (#584, #590, #599, #612, #620, #621).
A result I'm happy with, from the showcase: on a viscous shock problem, WENO on a grid clustered around the layer reached about two orders of magnitude lower error than a uniform grid with twice the points. Concentrating resolution where the physics needs it really pays off.
Everything above is merged and released. A user can now pass any monotone grid vector and use WENO end to end:
disc = MOLFiniteDifference([x => xgrid], t; advection_scheme = WENOScheme())
prob = discretize(pdesys, disc)Some limitations are documented in the tutorial: the formulation is non-conservative, mismatched-grid interfaces currently support first-order derivatives only, and the advection velocity needs to be continuous across an interface.
The GSoC scope itself is complete. The natural next step is grid adaptivity — placing points well currently requires knowing where the steep features are in advance. I opened issue #610 with a roadmap for this, and it's something I'd like to keep working on with the SciML community after GSoC. The limitations listed above are also good candidates for follow-up contributions.
All work was merged into SciML/MethodOfLines.jl:
| PR | Description |
|---|---|
| #562 | Non-uniform regression tests for the upwind scheme |
| #542 | Dispatch architecture for non-uniform WENO |
| #581 | Non-uniform WENO-5 core |
| #582 | Boundary conditions for non-uniform WENO |
| #583 | Discretizer integration |
| #591 | MMS convergence tests |
| #602 | Interface and periodic boundaries for non-uniform WENO |
| #572 | Interface and periodic boundaries for non-uniform upwind |
| #609 | Benchmark suite (runs on every PR) |
| #615 | Tutorial |
| #616 | Quantitative showcase |
| #584, #590, #599, #612, #620, #621 | Various CI and compatibility fixes |
The math came before the code. The standard WENO formulas quietly assume uniform spacing, so I spent a good part of the summer deriving and verifying the non-uniform versions before writing any integration code. Testing against exact solutions from day one saved me many times.
Small, stacked PRs work. Splitting the project into reviewable pieces made feedback faster and kept the uniform path verifiably untouched at every step.
Keeping CI green is part of the job. When upstream changes broke tests mid-project, fixing them wasn't a distraction from the project — it was what made reviewing the project possible. That gave me a bit of a maintainer's perspective I didn't have before.
Measure honestly. One of the more useful benchmark results was a negative one: a stretched grid on its own doesn't help unless it's actually adapted to the solution. That observation is what shaped the adaptivity roadmap.
GSoC 2026 series
Final Work Product (this post)