SimiLie

Simulation in Lie groups.

This is a C++ performance-portable project, supporting CPU & GPU. For simplicity sake and performance purpose (avoid access to adjacency matrix, efficient parallelization & memory access), only mesh structured as a cartesian product of 1D meshes is supported. No adaptative mesh refinement. Main focus are modularity, versatility and performance.

Relies on DDC.

Absolute WIP, there is no guarantee for proper working.

Relies on the #embed directive from C23. As it is not yet in C++ (even in C++26), the only supported compiler seems to be Clang 19.

Development plan

1. Short term (month): simulate electromagnetism on manifold.
2. Mid term (months): simulate geometrodynamics.
3. Long term: generalize to all kinds of classical gauge theories and put Lie groups at the center of the construction.
4. Extra-long term (years): quantization.

Initial roadmap

DONE:

• Create repository.
• Initial setup (dependencies, basic test, project structure).
• Simple mesher.
• Tensors.

TODO:

• Differential forms & fields of differential forms.
• Basic useful differential forms (not involving derivatives): tetrads, metric.
• More advanced tools: discrete Hodge star, exterior derivative, codifferential operator, covariant derivative.
• More advanced useful differential forms: connections, Riemann, Ricci.
• Integration, inner product.
• Weak formulation, finite element method applied to electrostatic problem on manifold.
• Generalization to electromagnetism.
• ...

References

• Gravitation. By C.W. Misner, K.S. Thorne and J.A. Wheeler. ISBN 0-7167-0334-3, ISBN 0-7167-0344-0 (pbk). San Francisco: W.H. Freeman and Company, 1973
• Desbrun, M., Kanso, E., Tong, Y. (2008). Discrete Differential Forms for Computational Modeling. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds) Discrete Differential Geometry. Oberwolfach Seminars, vol 38. Birkhäuser Basel.
• Finite element exterior calculus. Douglas N. Arnold. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM 2018.
• Celada, M., González, D., & Montesinos, M. (2016). BF gravity. Classical and Quantum Gravity, 33(21), 213001.
• Group Theory, Birdtracks, Lie’s, and Exceptional Groups, Predrag Cvitanović, Princeton University Press July 2008.