Project B9 • Dynamical low-rank approximation for the simulation of radiation heat waves

Principal investigators

Project summary

This project aims at applying, adapting and implementing the dynamical low-rank method for the simulation of radiation heat waves. We combine the dynamical low-rank method with a high-order Discontinuous Galerkin (DG) discretization in space, and either a spectral or collocation method in angle. The ansatz functions are tensorized so that one can interpret the semi-discretization in space, angle and frequency as a tensor-valued ordinary differential equation in time. The dynamical low-rank approximation is a low-rank factorization updating technique. It leads to differential equations for factors in a decomposition of the solution, which need to be solved numerically. The dynamical low-rank method seems particularly suitable for our purposes, because in many relevant test cases the propagation of radiation heat waves can be described by an asymptotic limit equation. Thus the effective dynamics takes place on a lower-dimensional manifold. In this way, the six-dimensional (3 space, 2 angle, 1 frequency) radiation transport problem is reduced, both in computational cost as well as in memory footprint.

The first step is to put this method into practice, using a DG discretization in space, a spectral method in angle and a finite volume method in frequency. It is important to construct the projected right hand side of the equation in an efficient way. On the method-oriented side, we develop modifications of the dynamical low-rank method to enforce the numerical scheme's conservation of the overall energy and mass of particles. Our strategy to achieve this is to introduce a Lagrange parameter to enforce conservation. The largest challenge in this project will be to make the numerical scheme uniformly stable in terms of the mean free path of the photons. Such a scheme is called asymptotic-preserving in the literature. To this end, certain substeps of the method use an implicit time discretization, and we control the numerical stabilization parameter. Finally, we investigate parallelization strategies so that we obtain an efficient high-performance computing (HPC) implementation. Together with application scientists, we perform simulations of radiation heat waves in realistic application test cases and especially investigate the memory use of the new method.

Publications

1. G. Ceruti, C. Lubich, and H. Walach. Time integration of tree tensor networks. SIAM J. Numer. Anal., 59(1):289–313, February 2021. URL https://doi.org/10.1137/20M1321838. [preprint] [bibtex]

2. G. Ceruti and C. Lubich. Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors. BIT, 1–24, January 2020. URL https://doi.org/10.1007/s10543-019-00799-8. Online first. [preprint] [bibtex]

Preprints

3. J. Kusch, L. Einkemmer, and G. Ceruti. On the stability of robust dynamical low-rank approximations for hyperbolic problems. CRC 1173 Preprint 2021/33, Karlsruhe Institute of Technology, July 2021. [files] [bibtex]

4. J. Kusch, G. Ceruti, L. Einkemmer, and M. Frank. Dynamical low-rank approximation for Burgers' equation with uncertainty. CRC 1173 Preprint 2021/20, Karlsruhe Institute of Technology, May 2021. [files] [bibtex]

5. G. Ceruti, J. Kusch, and C. Lubich. A rank-adaptive robust integrator for dynamical low-rank approximation. CRC 1173 Preprint 2021/16, Karlsruhe Institute of Technology, April 2021. [bibtex]

6. G. Ceruti and C. Lubich. An unconventional robust integrator for dynamical low-rank approximation. CRC 1173 Preprint 2020/41, Karlsruhe Institute of Technology, December 2020. Accepted by BIT Numerical Mathematics. [bibtex]

Theses

7. J. Kusch. Realizability-preserving discretization strategies for hyperbolic and kinetic equations with uncertainty. PhD thesis, Karlsruhe Institute of Technology, May 2020. [bibtex]