2000 character limit reached
Decapodes: A Diagrammatic Tool for Representing, Composing, and Computing Spatialized Partial Differential Equations
Published 30 Jan 2024 in math.NA and cs.NA | (2401.17432v1)
Abstract: We present Decapodes, a diagrammatic tool for representing, composing, and solving partial differential equations. Decapodes provides an intuitive diagrammatic representation of the relationships between variables in a system of equations, a method for composing systems of partial differential equations using an operad of wiring diagrams, and an algorithm for deriving solvers using hypergraphs and string diagrams. The string diagrams are in turn compiled into executable programs using the techniques of categorical data migration, graph traversal, and the discrete exterior calculus. The generated solvers produce numerical solutions consistent with state-of-the-art open source tools as demonstrated by benchmark comparisons with SU2. These numerical experiments demonstrate the feasibility of this approach to multiphysics simulation and identify areas requiring further development.
- Ralph Abraham, Jerrold E. Marsden and Tudor Ratiu “Manifolds, tensor analysis, and applications” Springer, 1988 DOI: 10.1007/978-1-4612-1029-0
- “Unified Form Language: A Domain-Specific Language for Weak Formulations of Partial Differential Equations” In ACM Trans. Math. Softw. 40.2 New York, NY, USA: Association for Computing Machinery, 2014 DOI: 10.1145/2566630
- Douglas N. Arnold “Finite Element Exterior Calculus” Philadelphia, PA: Society for IndustrialApplied Mathematics, 2018 DOI: 10.1137/1.9781611975543
- Rama Ayoub, Aziz Hamdouni and Dina Razafindralandy “A new Hodge operator in discrete exterior calculus. Application to fluid mechanics” Company: Communications on Pure & Applied Analysis Distributor: Communications on Pure & Applied Analysis Institution: Communications on Pure & Applied Analysis Label: Communications on Pure & Applied Analysis Publisher: American Institute of Mathematical Sciences In Communications on Pure & Applied Analysis 20.6, 2021, pp. 2155 DOI: 10.3934/cpaa.2021062
- John C. Baez, Kenny Courser and Christina Vasilakopoulou “Structured versus Decorated Cospans” In Compositionality 4 Compositionality, 2022, pp. 3 DOI: 10.32408/compositionality-4-3
- John C. Baez and Brendan Fong “A Compositional Framework for Passive Linear Networks” arXiv: 1504.05625 In arXiv:1504.05625 [math-ph], 2018 URL: http://arxiv.org/abs/1504.05625
- John C. Baez and Blake S. Pollard “A Compositional Framework for Reaction Networks” arXiv: 1704.02051 In Reviews in Mathematical Physics 29.09, 2017, pp. 1750028 DOI: 10.1142/S0129055X17500283
- “Rewriting modulo Symmetric Monoidal Structure” In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16 New York, NY, USA: Association for Computing Machinery, 2016, pp. 710–719 DOI: 10.1145/2933575.2935316
- “String Diagram Rewrite Theory I: Rewriting with Frobenius Structure” In J. ACM 69.2, 2022, pp. 14:1–14:58 DOI: 10.1145/3502719
- Alain Bossavit “On the geometry of electromagnetism (4): Maxwell’s house” In J. Japan Soc. Appl. Electromagn. & Mech 6.4, 1998, pp. 12–20
- Philip Boyland “Fluid mechanics and mathematical structures” In An introduction to the geometry and topology of fluid flows Springer, 2001, pp. 105–134 DOI: 10.1007/978-94-010-0446-6˙6
- Andrea Censi “Monotone co-design problems; or, everything is the same” In 2016 American Control Conference (ACC), 2016, pp. 1227–1234 DOI: 10.1109/ACC.2016.7525085
- Keenan Crane “Discrete Differential Geometry: An Applied Introduction”, 2022, pp. 173
- “Discrete Exterior Calculus” arXiv:math/0508341 type: article, 2005 URL: http://arxiv.org/abs/math/0508341
- Georges A. Deschamps “Electromagnetics and differential forms” In Proceedings of the IEEE 69.6, 1981, pp. 676–696 DOI: 10.1109/PROC.1981.12048
- “SU2: An Open-Source Suite for Multiphysics Simulation and Design” In AIAA Journal 54.3, 2016, pp. 828–846 DOI: 10.2514/1.J053813
- Conal Elliott “Compiling to Categories” In Proc. ACM Program. Lang. 1.ICFP New York, NY, USA: Association for Computing Machinery, 2017 DOI: 10.1145/3110271
- Brendan Fong and David I. Spivak “Seven Sketches in Compositionality: An Invitation to Applied Category Theory” arXiv: 1803.05316 In arXiv:1803.05316 [math], 2018 URL: http://arxiv.org/abs/1803.05316
- Anil N. Hirani “Discrete exterior calculus”, 2003
- “A primitive variable discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes” In Physics of Fluids 33.1, 2021, pp. 017114 DOI: 10.1063/5.0035981
- “Multiphysics simulations: Challenges and opportunities” In The International Journal of High Performance Computing Applications 27.1, 2013, pp. 4–83 DOI: 10.1177/1094342012468181
- Anders Logg, Garth Wells and Kent-Andre Mardal “Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book” In Lecture Notes in Computational Science and Engineering 84, 2011 DOI: 10.1007/978-3-642-23099-8
- Mamdouh Mohamed, Anil N. Hirani and Ravi Samtaney “Numerical convergence of discrete exterior calculus on arbitrary surface meshes” In International Journal for Computational Methods in Engineering Science and Mechanics 19, 2018, pp. 194–206
- Mamdouh S. Mohamed, Anil N. Hirani and Ravi Samtaney “Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes” In Journal of Computational Physics 312 Elsevier BV, 2016, pp. 175–191 DOI: 10.1016/j.jcp.2016.02.028
- Evan Patterson, Owen Lynch and James Fairbanks “Categorical Data Structures for Technical Computing” arXiv:2106.04703 [cs, math], 2022 DOI: 10.32408/compositionality-4-5
- “A diagrammatic view of differential equations in physics” In Mathematics in Engineering 5.2, 2023, pp. 1–59 DOI: 10.3934/mine.2023036
- “A diagrammatic view of differential equations in physics” Cc_license_type: cc_by Number: mine-05-02-036 Primary_atype: Mathematics in Engineering Subject_term: Research article Subject_term_id: Research article In Mathematics in Engineering 5.2, 2023, pp. 1–59 DOI: 10.3934/mine.2023036
- “Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia” In Journal of Open Research Software 5.1 Ubiquity Press, 2017, pp. 15
- “Firedrake: Automating the Finite Element Method by Composing Abstractions” In ACM Trans. Math. Softw. 43.3 New York, NY, USA: Association for Computing Machinery, 2016 DOI: 10.1145/2998441
- Marko Šešlija, Jacquelien M.A. Scherpen and Arjan Schaft “A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems” In 2011 50th IEEE Conference on Decision and Control and European Control Conference, 2011, pp. 7003–7008 DOI: 10.1109/CDC.2011.6160579
- Peter M. Sockol “Multigrid solution of the Navier-Stokes equations at low speeds with large temperature variations” In Journal of Computational Physics 192, 2003, pp. 570–592
- David I. Spivak “Database queries and constraints via lifting problems” In Mathematical Structures in Computer Science 24.6, 2014 DOI: 10.1017/S0960129513000479
- Enzo Tonti “On the mathematical structure of a large class of physical theories” In Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 52.1, 1972, pp. 48–56 URL: https://eudml.org/doc/295907
- Enzo Tonti “The Mathematical Structure of Classical and Relativistic Physics”, Modeling and Simulation in Science, Engineering and Technology New York, NY: Springer New York, 2013 DOI: 10.1007/978-1-4614-7422-7
- Scott O. Wilson “Differential forms, fluids, and finite models” In Proceedings of the American Mathematical Society 139.7, 2011, pp. 2597–2604 DOI: 10.1090/S0002-9939-2011-11003-7
Paper Prompts
Sign up for free to create and run prompts on this paper.