Dynamical low-rank integrators for second-order matrix differential equations
Tóm tắt
In this paper, we construct and analyze a new dynamical low-rank integrator for second-order matrix differential equations. The method is based on a combination of the projector-splitting integrator introduced in Lubich and Oseledets (BIT 54(1):171–188, 2014.
https://doi.org/10.1007/s10543-013-0454-0
) and a Strang splitting. We also present a variant of the new integrator which is tailored to semilinear second-order problems.
Tài liệu tham khảo
Carle, C., Hochbruck, M., Sturm, A.: On leapfrog-Chebyshev schemes. SIAM J. Numer. Anal. 58(4), 2404–2433 (2020). https://doi.org/10.1137/18M1209453
Ceruti, G., Lubich, C.: An unconventional robust integrator for dynamical low-rank approximation. BIT 62(1), 23–44 (2022). https://doi.org/10.1007/s10543-021-00873-0
Ceruti, G., Lubich, C., Walach, H.: Time integration of tree tensor networks. SIAM J. Numer. Anal. 59(1), 289–313 (2021). https://doi.org/10.1137/20M1321838
Einkemmer, L., Lubich, C.: A low-rank projector-splitting integrator for the Vlasov–Poisson equation. SIAM J. Sci. Comput. 40(5), B1330–B1360 (2018). https://doi.org/10.1137/18M116383X
Einkemmer, L., Ostermann, A., Piazzola, C.: A low-rank projector-splitting integrator for the Vlasov–Maxwell equations with divergence correction. J. Comput. Phys. 403, 109,063 (2020)
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer. 12, 399–450 (2003). https://doi.org/10.1017/S0962492902000144
Hochbruck, M., Neher, M., Schrammer, S.: Dynamical low-rank integrators for second-order matrix differential equations. CRC 1173 Preprint 2022/12, Karlsruhe Institute of Technology (2022). https://doi.org/10.5445/IR/1000143198. https://www.waves.kit.edu/downloads/CRC1173_Preprint_2022-13.pdf
Joly, P., Rodríguez, J.: Optimized higher order time discretization of second order hyperbolic problems: construction and numerical study. J. Comput. Appl. Math. 234(6), 1953–1961 (2010). https://doi.org/10.1016/j.cam.2009.08.046
Kieri, E., Lubich, C., Walach, H.: Discretized dynamical low-rank approximation in the presence of small singular values. SIAM J. Numer. Anal. 54(2), 1020–1038 (2016). https://doi.org/10.1137/15M1026791
Koch, O., Lubich, C.: Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29(2), 434–454 (2007). https://doi.org/10.1137/050639703
Kusch, J., Ceruti, G., Einkemmer, L., Frank, M.: Dynamical low-rank approximation for Burgers’ equation with uncertainty. Int. J. Uncertain. Quantif. 12(5), 1–21 (2022). https://doi.org/10.1615/int.j.uncertaintyquantification.2022039345
Lubich, C., Oseledets, I.V.: A projector-splitting integrator for dynamical low-rank approximation. BIT 54(1), 171–188 (2014). https://doi.org/10.1007/s10543-013-0454-0
Lubich, C., Oseledets, I.V., Vandereycken, B.: Time integration of tensor trains. SIAM J. Numer. Anal. 53(2), 917–941 (2015). https://doi.org/10.1137/140976546
Lubich, C., Vandereycken, B., Walach, H.: Time integration of rank-constrained Tucker tensors. SIAM J. Numer. Anal. 56(3), 1273–1290 (2018). https://doi.org/10.1137/17M1146889
Ostermann, A., Piazzola, C., Walach, H.: Convergence of a low-rank Lie–Trotter splitting for stiff matrix differential equations. SIAM J. Numer. Anal. 57(4), 1947–1966 (2019). https://doi.org/10.1137/18M1177901
Schrammer, S.: On dynamical low-rank integrators for matrix differential equations. Ph.D. thesis, Karlsruher Institut für Technologie (KIT) (2022). https://doi.org/10.5445/IR/1000148853