Schroedinger's Smoke

Albert Chern, Felix Knöppel, Ulrich Pinkall, Peter Schröder, Steffen Weißmann



A new approach for the purely Eulerian simulation of incompressible fluids is described here. [1] In it, the fluid state is represented by a $\mathbb{C}^2$- valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type.

The latter ensures that dynamics due to thin vortical structures, all important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids.

Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity.

Its use is demonstrated in a variety of scenarios, compared with experiments, and evaluated against benchmark tests.

A full implementation is found on the project site .


  • Albert Chern, Felix Knöppel, Peter Pinkall, Ulrich and​ ​Schröder, and Steffen Weißmann.
    Schrödinger's Smoke.
    ACM Trans. Graph., 35:77:1–77:13, July 2016.

Dr. Albert Chern   +

University: TU Berlin
E-Mail: chern[at]

Dr. Felix Knöppel   +

Projects: C07
University: TU Berlin
E-Mail: knoeppel[at]

Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin, Institut für Mathematik, MA 822
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424607
E-Mail: pinkall[at]

Prof. Dr. Peter Schröder   +