## Seminar -

Geometric formulations of inviscid fluids and their discretization

- OvGU - 2019

Lecturer: Jan Heiland, Christian Lessig

This seminar in the LSF

Day | Time | Place |
---|---|---|

Wednesday | 11:00-13:00 | G05-122 |

Consultation hours: Please make an appointment by email.

## Topics

### Variational formulation of ideal fluid

Develop the variational formulation of ideal fluid dynamics, possibly including the general perspective of Euler-Poincare reduction.

V. I. Arnold, *Mathematical Methods of Classical Mechanics*, Second. ed. Springer, 1989 (Appendix A).

J. E. Marsden and T. S. Ratiu, *Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems*, Third ed. New York: Springer-Verlag, 1999.

### Hamiltonian formulation of ideal fluid

Develop the Hamiltonian formulation of ideal fluid dynamics, including conserved quantities in 2D and 3D.

J. E. Marsden and A. Weinstein, *Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,* Phys. D Nonlinear Phenom., vol. 7, no. 1–3, pp. 305–323, May 1983. [pdf]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics. New York: Springer, 1998.

### Algebraic, structure preserving numerical method for ideal fluid dynamics

Theory: Develop the discretization of 2D ideal fluid dynamics proposed by Zeitlin based on the work by Hoppe.

Implementation: Develop an implementation of the algorithm proposed by Zeitlin, as done by McLachlan.

V. Y. Zeitlin, *Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure,* Phys. D Nonlinear Phenom., vol. 49, no. 3, pp. 353–362, Apr. 1991.

J. Hoppe, *Diffeomorphism Groups, Quantization, and SU(∞),* Int. J. Mod. Phys. A, vol. 04, no. 19, pp. 5235–5248, Nov. 1989.

S. J. Rankin, *SU(∞) and the large-N limit,* Ann. Phys. (N. Y)., vol. 218, no. 1, pp. 14–50, Aug. 1992.

R. I. McLachlan, *Explicit Lie-Poisson integration and the Euler equations,* Phys. Rev. Lett., vol. 71, no. 19, pp. 3043–3046, Nov. 1993. [pdf]

### Madelung transform

Develop the Madelung transform as a hydrodynamical model for the Schrödinger equation, with an emphasis on the geometric perspective as a momentum map that connects it to compressible fluid dynamics.

Optional: Also develop the connection to the work by [Chern et al. 2016].

D. Fusca, *The Madelung transform as a momentum map,* J. Geom. Mech., vol. 9, no. 2, pp. 157–165, 2017. [pdf]

E. Madelung, *Quantentheorie in hydrodynamischer Form,* Zeitschrift für Phys., vol. 40, no. 3–4, pp. 322–326, Mar. 1927.

E. Madelung, *Eine anschauliche Deutung der Gleichung von Schrödinger,* Naturwissenschaften, vol. 14, no. 45, pp. 1004–1004, Nov. 1926.

A. Chern, F. Knöppel, U. Pinkall, P. Schröder, and S. Weißmann, *Schrödinger’s smoke,* ACM Trans. Graph., vol. 35, no. 4, pp. 1–13, Jul. 2016.

M. Schönberg, *On the hydrodynamical model of the quantum mechanics,* Nuovo Cim., vol. 12, no. 1, pp. 103–133, Jul. 1954.

### Spectral, structure preserving integrator for ideal fluid dynamics

Develop an implementation of the algorithm proposed by Liu et al. [2015].

B. Liu, G. Mason, J. Hodgson, Y. Tong, and M. Desbrun, *Model-reduced variational fluid simulation*, ACM Trans. Graph., vol. 34, no. 6, pp. 1–12, Oct. 2015.

J. E. Marsden and A. Weinstein, *Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids,* Phys. D Nonlinear Phenom., vol. 7, no. 1–3, pp. 305–323, May 1983. [pdf]

T. de Witt, C. Lessig, and E. Fiume, *Fluid Simulation Using Laplacian Eigenfunctions,* ACM Trans. Graph., vol. 31, no. 1, pp. 1–11, Jan. 2012. [pdf]

### Variational, structure preserving numerical integrator for arbitrary meshes

Theory: Develop the variational structure preserving integrator proposed by Pavlov et al. [2011], possibly including the extensions in Gawlik et al. [2011].

Implementation: Implement the algorithm presented by Mullen et al. (2D, on a regular grid or using PyDec).

E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, and M. Desbrun, *Geometric, variational discretization of continuum theories,* Phys. D Nonlinear Phenom., vol. 240, no. 21, pp. 1724–1760, Oct. 2011. [pdf]

D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden, and M. Desbrun, *Structure-preserving discretization of incompressible fluids,* Phys. D Nonlinear Phenom., vol. 240, no. 6, pp. 443–458, Mar. 2011.

P. Mullen, K. Crane, D. Pavlov, Y. Tong, and M. Desbrun, *Energy-Preserving Integrators for Fluid Animation,* ACM Trans. Graph. (Proceedings SIGGRAPH 2009), vol. 28, no. 3, pp. 1–8, 2009.

### Vorticity-based, structure preserving discretization of fluids

Develop an implementation of the technique proposed by Elcott et al. [2007]

S. Elcott, Y. Tong, E. Kanso, P. Schröder, and M. Desbrun, *Stable, Circulation-Preserving, Simplicial Fluids,* ACM Trans. Graph., vol. 26, no. 1, 2007. [pdf]

## General literature

V. I. Arnold, *Mathematical Methods of Classical Mechanics*, Second. ed. Springer, 1989.

J. E. Marsden, Lectures on Mechanics, Cambridge University Press, 1992.

J. E. Marsden and T. S. Ratiu, *Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems*, third ed., New York: Springer-Verlag, 1999.

D. D. Holm, T. Schmah, and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford University Press, 2009.

Tudor Ratiu. A Crash Course in Geometric Mechanics, third cycle. Monastir (Tunisie), 2005

R. Abraham and J. E. Marsden, Foundations of Mechanics, second ed., Addison-Wesley Publishing Company, Inc., 1978.

J. E. Marsden, T. S. Ratiu, and R. Abraham, *Manifolds, Tensor Analysis, and Applications*, Third ed., New York: Springer-Verlag, 2004.

E. Hairer, C. Lubich, and G. Wanner, *Geometric Numerical Integration*, Second ed. Springer-Verlag, 2006.

J. E. Marsden and M. West, *Discrete Mechanics and Variational Integrators*, Acta Numer., vol. 10, pp. 357–515, 2001.

A. Stern and M. Desbrun, Discrete Geometric Mechanics for Variational Time Integrators, in SIGGRAPH ’06: ACM SIGGRAPH 2006 Courses., 2006.