Research References: Numerical Methods and Geometric Discretisation

Numerical Integration of Differential Equations

• Ernst Hairer, Syvert P. Nørsett and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 1993. (eBook)

• Ernst Hairer and Gerhard Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 1996. (eBook)

• Ernst Hairer, Christian Lubich, Michel Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer, 1989. (eBook)

• Peter Deuflhard, Folkmar Bornemann. Scientific Computing with Ordinary Differential Equations. Springer, 2002. (eBook)

• John C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, 2016. (eBook)

Geometric Numerical Integration

• Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration Illustrated by the Störmer–Verlet Method. Acta Numerica 12, 399-450, 2003. (Journal)

• Ernst Hairer, Christian Lubich and Gerhard Wanner. Geometric Numerical Integration. Springer, 2006. (eBook)

• Sergio Blanes, Fernando Casas. A Concise Introduction to Geometric Numerical Integration. CRC Press, 2016. (eBook)

• Benedict Leimkuhler and Sebastian Reich. Simulating Hamiltonian Dynamics. Cambridge University Press, 2005. (eBook)

• Jesús Maria Sanz-Serna, Manuel P. Calvo. Numerical Hamiltonian Problems. Chapman Hall, 1994.

Discrete Mechanics and Field Theory, Variational and Multisymplectic Integrators

• Jerrold E. Marsden, George W. Patrick and Steve Shkoller: Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs (1998)
  Communications in Mathematical Physics, Vol. 199, p. 351-395, arXiv:math/9807080

• Jerrold E. Marsden and Matthew West: Discrete Mechanics and Variational Integrators (2001)
  Acta Numerica 2001, p. 357-514

• Mathieu Desbrun, Anil N. Hirani, Melvin Leok and Jerrold E. Marsden: Discrete Exterior Calculus (2005)
  arXiv:math/0508341

• Ari Stern, Yiying Tong, Mathieu Desbrun and Jerrold E. Marsden: Geometric Computational Electrodynamics (2007)
  arXiv:0707.4470

• Tomasz M. Tyranowski, Mathieu Desbrun: Variational Partitioned Runge-Kutta Methods for Lagrangians Linear in Velocities (2014), arXiv:1401.7904

Discrete Euler-Poincaré Reduciton

• Evan S. Gawlik, Patrick Mullen, Dmitry Pavlov, Jerrold E. Marsden and Mathieu Desbrun: Geometric, Variational Discretization of Continuum Theories (2011)
  Physica D: Nonlinear Phenomena, Vol. 240, p. 1724–1760, arXiv:1010.4851

• Dmitry Pavlov, Patrick Mullen, Yiying Tong, Eva Kanso, Jerrold E. Marsden, Mathieu Desbrun: Structure-Preserving Discretization of Incompressible Fluids (2011)
  Physica D, Vol. 240, p. 443–458, arXiv:0912.3989

Theses

• Anil N. Hirani: Discrete Exterior Calculus (PhD Thesis, 2003)
  Caltech Thesis

• Melvin Leok: Foundations of Computational Geometric Mechanics (PhD Thesis, 2004)
  Caltech Thesis

• Matthew West: Variational Integrators (PhD Thesis, 2004)
  Caltech Thesis

• Ari Stern: Geometric Discretization of Lagrangian Mechanics and Field Theories (PhD Thesis, 2009)
  Caltech Thesis

• Dmitry Pavlov: Structure-Preserving Discretization of Incompressible Fluids (PhD Thesis, 2009)
  Caltech Thesis

• Patrick G. Mullen: Eulerian Geometric Discretizations of Manifolds and Dynamics (PhD Thesis, 2012)
  Caltech Thesis