rrq

tools for high‑order quaternionic product quadratures that resolve singular and nearly singular 3D Laplace layer potentials. The implementation follows Ref. 1 and Ref. 2.

Layout

• src/ and matlab/: maintained Fortran RRQ implementation and MATLAB MEX gateway.
• utils/: maintained MATLAB quadrature helpers.
• external/: LineQuaaadrature, QuatApproximation, and kd-tree submodules.
• test/: maintained native and MATLAB stellarator tests, with surface meshes generated in memory by the LineQuaaadrature Fortran/MEX geometry interfaces.
• web/stellarator/: browser solver and its build and test pipeline.
• archive/legacy-code/: inactive historical MATLAB demos and kernel helpers; this directory is not added by setup.m.

Build and benchmark

git submodule update --init --recursive
make lib
make -C test
OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 ./build/stellarator_bench 5 128 0

This runs orders 4–12 with the 128-wide SIMD kernel and uniform refinement.

The torus comparison additionally requires a built FMM3DBIE checkout.

make fmm3dbie-mex FMM3DBIE_DIR=~/git/fmm3dbie

Compile the platform-specific kdtree MEX files once from MATLAB:

cd('/path/to/qotential/external/kdtree/toolbox')
kdtree_compile

References

1. Hai Zhu, and Shravan Veerapaneni. 2022. “High-Order Close Evaluation of Laplace Layer Potentials: A Differential Geometric Approach.” SIAM Journal on Scientific Computing.
2. Shidong Jiang, and Hai Zhu. 2024. “Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions.” arXiv preprint arXiv:2411.08342.

Supported kernels

• A set of kernels: $\mathcal{K}=\lbrace\Delta u=0;\ (\Delta+k^2)u=0;\ \mu\Delta\mathbf{u}-\nabla p=0,\ \nabla \cdot\mathbf{u}=0\rbrace\times\lbrace\mathcal{S},\mathcal{D},\mathcal{S}',\mathcal{D}'\rbrace$

• rrq: $\mathcal{K}_{\mathrm{rrq}}=\lbrace\Delta u=0\rbrace\times\lbrace\mathcal{S},\mathcal{D}\rbrace\subsetneq\mathcal{K}$

• wish list: $\mathcal{K}\setminus\mathcal{K}_{\mathrm{rrq}}$ and complexification

Development is ongoing and the wish-list kernels will certainly be implemented if they can be done, though no timeline is fixed given my limited bandwidth.

There are three main components, and I would appreciate help on any of them: kernel-splitting, line quadrature, and quaternion approximation.

The underlying ideas are largely intuitive, but the implementation details can be a bit involved unfortunately. Some rely on established literature, some introduce new ingredients, some are not used off the shelf but adapted to the specific needs of rrq, and a few still require fixes, which are planned. For example, the local-to-local translation operator is adapted from FMM3D, and the singularity-swap line quadrature is adaptived from the linequad.

To do list

• (to do) solid angle fix
• (to do) check examples on different systems, and update readme
• (to do) minimal Helmholtz slp + dlp
• (to do) minimal Stokes slp...
• (to do) check, and move all line quadrature to LQ submodule, move all quaternion approximation and locloc related to QA
• (to do) quaternion approximation on curved element (thm)
• (to do) laplace slpn, dlpn, maybe (no legacy code available...) slpnn, dlpnn
• (to do) helmholtz slpn + dlpn
• (to do) stokes (no legacy code available...) slpn, dlp, dlpn...