Literature#
Zoomy implements shallow-water, shallow-moment and vertically averaged models, finite-volume reconstruction, flux, time-stepping and projection methods, and linear and nonlinear solvers; this page lists the publications they follow, and the docstrings of the implementing functions cite these entries by DOI. The software Zoomy builds on is credited on the Acknowledgements page.
Ann S. Almgren, John B. Bell, and William Y. Crutchfield. Approximate Projection Methods: Part I. Inviscid Analysis. SIAM Journal on Scientific Computing, 22(4):1139–1159, 2000. doi:10.1137/S1064827599357024.
Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau, Rupert Klein, and Benoît Perthame. A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows. SIAM Journal on Scientific Computing, 25(6):2050–2065, 2004. doi:10.1137/S1064827503431090.
Emmanuel Audusse and Marie-Odile Bristeau. A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. Journal of Computational Physics, 206(1):311–333, 2005. doi:10.1016/j.jcp.2004.12.016.
Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, and Jacques Sainte-Marie. A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM: Mathematical Modelling and Numerical Analysis, 45(1):169–200, 2011. doi:10.1051/m2an/2010036.
Timothy Barth and Dennis Jespersen. The design and application of upwind schemes on unstructured meshes. In 27th Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics, 1989. doi:10.2514/6.1989-366.
Nathan Bell, Luke N. Olson, and Jacob Schroder. PyAMG: Algebraic Multigrid Solvers in Python. Journal of Open Source Software, 7(72):4142, 2022. doi:10.21105/joss.04142.
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Alfredo Bermudez and Ma Elena Vazquez. Upwind methods for hyperbolic conservation laws with source terms. Computers & Fluids, 23(8):1049–1071, 1994. doi:10.1016/0045-7930(94)90004-3.
James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Yash Katariya, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. JAX: composable transformations of Python+NumPy programs. jax-ml/jax, 2018.
F. N. Cantero-Chinchilla, O. Castro-Orgaz, and A. A. Khan. Depth-integrated nonhydrostatic free-surface flow modeling using weighted-averaged equations. International Journal for Numerical Methods in Fluids, 87(1):27–50, 2018. doi:10.1002/fld.4481.
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Gerasimos Chourdakis, Kyle Davis, Benjamin Rodenberg, Miriam Schulte, Frédéric Simonis, Benjamin Uekermann, Georg Abrams, Hans-Joachim Bungartz, Lucia Cheung Yau, Ishaan Desai, Konrad Eder, Richard Hertrich, Florian Lindner, Alexander Rusch, Dmytro Sashko, David Schneider, Amin Totounferoush, Dominik Volland, Peter Vollmer, and Oguz Ziya Koseomur. preCICE v2: A sustainable and user-friendly coupling library. Open Research Europe, 2:51, 2022. doi:10.12688/openreseurope.14445.2.
S. Clain, S. Diot, and R. Loubère. A high-order finite volume method for systems of conservation laws—Multi-dimensional Optimal Order Detection (MOOD). Journal of Computational Physics, 230(10):4028–4050, 2011. doi:10.1016/j.jcp.2011.02.026.
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C. Escalante, M. Dumbser, and M. J. Castro. An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes. Journal of Computational Physics, 394:385–416, 2019. doi:10.1016/j.jcp.2019.05.035.
C. Escalante, T. Morales de Luna, F. Cantero-Chinchilla, and O. Castro-Orgaz. Vertically averaged and moment equations: New derivation, efficient numerical solution and comparison with other physical approximations for modeling non-hydrostatic free surface flows. Journal of Computational Physics, 504:112882, 2024. doi:10.1016/j.jcp.2024.112882.
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin. Finite volume methods. In Philippe G. Ciarlet and Jacques-Louis Lions, editors, Handbook of Numerical Analysis, Volume 7: Techniques of Scientific Computing (Part 3), pages 713–1018. Elsevier, 2000. doi:10.1016/s1570-8659(00)07005-8.
Robert D. Falgout and Ulrike Meier Yang. hypre: A Library of High Performance Preconditioners, pages 632–641. Springer Berlin Heidelberg, 2002. doi:10.1007/3-540-47789-6_66.
J. Fe, F. Navarrina, J. Puertas, P. Vellando, and D. Ruiz. Experimental validation of two depth-averaged turbulence models. International Journal for Numerical Methods in Fluids, 60(2):177–202, 2009. doi:10.1002/fld.1880.
Jean-Frédéric Gerbeau and Benoit Perthame. Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation. Discrete and Continuous Dynamical Systems - B, 1(1):89–102, 2001. doi:10.3934/dcdsb.2001.1.89.
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Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor. Strong Stability-Preserving High-Order Time Discretization Methods. SIAM Review, 43(1):89–112, 2001. doi:10.1137/S003614450036757X.
Andreas Griewank. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation. Optimization Methods and Software, 1(1):35–54, 1992. doi:10.1080/10556789208805505.
Andreas Griewank and Andrea Walther. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. Society for Industrial and Applied Mathematics, 2008. ISBN 9780898717761. doi:10.1137/1.9780898717761.
Amiram Harten, Peter D. Lax, and Bram van Leer. On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws. SIAM Review, 25(1):35–61, 1983. doi:10.1137/1025002.
Van Emden Henson and Ulrike Meier Yang. BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Applied Numerical Mathematics, 41(1):155–177, 2002. doi:10.1016/S0168-9274(01)00115-5.
Christopher A. Kennedy and Mark H. Carpenter. Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Applied Numerical Mathematics, 44(1-2):139–181, 2003. doi:10.1016/s0168-9274(02)00138-1.
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Julian Koellermeier and Marvin Rominger. Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations. Communications in Computational Physics, 28(3):1038–1084, 2020. doi:10.4208/cicp.oa-2019-0065.
Julia Kowalski and Manuel Torrilhon. Moment Approximations and Model Cascades for Shallow Flow. Communications in Computational Physics, 25(3):669–702, 2019. doi:10.4208/cicp.OA-2017-0263.
Alexander Kurganov and Guergana Petrova. A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System. Communications in Mathematical Sciences, 5(1):133–160, 2007. doi:10.4310/CMS.2007.v5.n1.a6.
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Randall J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002. ISBN 9780511791253. doi:10.1017/CBO9780511791253.
Yi Liu, Xiantao Fan, and Jian-Xun Wang. JAX-AMG: A GPU-accelerated differentiable sparse linear solver library for JAX. SoftwareX, 35:102966, 2026. doi:10.1016/j.softx.2026.102966.
Fabien Marche. Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European Journal of Mechanics - B/Fluids, 26(1):49–63, 2007. doi:10.1016/j.euromechflu.2006.04.007.
Dimitri Mavriplis. Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes. In 16th AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics, 2003. doi:10.2514/6.2003-3986.
M. Naumov, M. Arsaev, P. Castonguay, J. Cohen, J. Demouth, J. Eaton, S. Layton, N. Markovskiy, I. Reguly, N. Sakharnykh, V. Sellappan, and R. Strzodka. AmgX: A Library for GPU Accelerated Algebraic Multigrid and Preconditioned Iterative Methods. SIAM Journal on Scientific Computing, 37(5):S602–S626, 2015. doi:10.1137/140980260.
Lorenzo Pareschi and Giovanni Russo. Implicit–Explicit Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation. Journal of Scientific Computing, 25(1):129–155, 2005. doi:10.1007/s10915-004-4636-4.
Carlos Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44(1):300–321, 2006. doi:10.1137/050628052.
Ernesto Pimentel-García, Manuel J. Castro, Christophe Chalons, Tomás Morales de Luna, and Carlos Parés. In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension. Journal of Computational Physics, 459:111152, 2022. doi:10.1016/j.jcp.2022.111152.
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Ullika Scholz, Julia Kowalski, and Manuel Torrilhon. Dispersion in Shallow Moment Equations. Communications on Applied Mathematics and Computation, 6(4):2155–2195, 2023. doi:10.1007/s42967-023-00325-2.
Jie Shen. Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials. SIAM Journal on Scientific Computing, 15(6):1489–1505, 1994. doi:10.1137/0915089.
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Yulong Xing, Xiangxiong Zhang, and Chi-Wang Shu. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Advances in Water Resources, 33(12):1476–1493, 2010. doi:10.1016/j.advwatres.2010.08.005.
Weiqun Zhang, Ann Almgren, Vince Beckner, John Bell, Johannes Blaschke, Cy Chan, Marcus Day, Brian Friesen, Kevin Gott, Daniel Graves, Max Katz, Andrew Myers, Tan Nguyen, Andrew Nonaka, Michele Rosso, Samuel Williams, and Michael Zingale. AMReX: a framework for block-structured adaptive mesh refinement. Journal of Open Source Software, 4(37):1370, 2019. doi:10.21105/joss.01370.