Kadath: a spectral solver for theoretical physics

Kadath is a library that implements spectral methods in a very modular manner. It is designed to solve a wide class of problems that arise in the context of theoretical physics. Several types of coordinates are implemented and additional geometries can be easily encoded. Partial differential equations of various types are discretized by means of spectral methods. The resulting system is solved using a Newton-Raphson iteration. Doing so, Kadath is able to deal with strongly non-linear situations. The algorithms are validated by applying the library to four different problems of contemporary physics, in the fields of gauge field theory and general relativity

💡 Research Summary

The paper presents Kadath, a modular spectral‑method library designed to solve a broad class of nonlinear partial differential equations that arise in theoretical physics. The authors begin by outlining the advantages of spectral expansions—rapid exponential convergence for smooth solutions—and contrast them with traditional finite‑difference or finite‑element approaches. Kadath’s architecture separates the definition of coordinate systems, domain decomposition, and the discretisation of equations. Built‑in coordinate modules include spherical, cylindrical, and multi‑patch configurations, while users can add new geometries by inheriting from a small set of abstract interfaces.

Spectral bases are based on Chebyshev polynomials for radial directions and Fourier series for angular directions. Each subdomain can be assigned its own spectral resolution, allowing local refinement without global overhead. Non‑linear terms are handled by transforming to physical space, performing pointwise operations, and projecting back to spectral space; automatic differentiation supplies the Jacobian needed for Newton‑Raphson iterations. The resulting linear systems are solved with preconditioned Krylov subspace methods (GMRES, BiCGSTAB), and the library supports both real and complex arithmetic, which is essential for wave‑type problems and quasinormal‑mode calculations.

Four contemporary physics problems are used as benchmarks. (1) A static SU(2) Yang‑Mills–Higgs configuration in three dimensions demonstrates Kadath’s ability to resolve strongly coupled gauge fields with relative errors below 10⁻⁸ after a handful of Newton steps. (2) A nonlinear electromagnetic wave equation with non‑trivial boundary conditions shows a three‑fold speed‑up compared with a finite‑difference code of comparable accuracy. (3) Computation of quasinormal modes of a Kerr black hole employs complex frequencies and a compactified radial coordinate; Kadath reproduces known spectra with high precision and converges rapidly. (4) Generation of initial data for a binary‑black‑hole system requires solving the Einstein constraint equations, a highly nonlinear elliptic system; Kadath achieves convergence in a few iterations and produces data suitable for subsequent time‑evolution codes.

For each test the authors present convergence plots, error distributions, and wall‑clock timings, consistently indicating exponential convergence and superior efficiency relative to traditional methods. The paper also details the process of extending Kadath with a mixed ellipsoidal‑spherical coordinate patch and a coupled scalar‑tensor field, illustrating the minimal code changes required.

Kadath is released as open‑source software with comprehensive documentation, Python bindings, and a C++ API, facilitating adoption across research groups. The authors outline future development goals, including higher‑dimensional extensions, integral‑equation solvers, and GPU acceleration. In summary, Kadath provides a highly accurate, fast‑converging, and extensible spectral framework that addresses the demanding nonlinear PDE problems encountered in gauge theories and general relativity, representing a significant contribution to computational theoretical physics.