2000 character limit reached
Symmetric-conjugate splitting methods for evolution equations of parabolic type (2401.04196v1)
Published 8 Jan 2024 in math.NA, cs.NA, and physics.comp-ph
Abstract: The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schr\"odinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs.
- A. Bandrauk, H. Shen. Improved exponential split operator method for solving the time-dependent Schrödinger equation. Chem. Phys. Lett. 176 (1991) 428–432.
- W. Bao. Ground states and dynamics of multicomponent Bose–Einstein condensates. Multiscale Model. Simul. 2 (2004) 210–236.
- W. Bao, Q. Du. Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674–1697.
- S. Blanes, F. Casas. On the necessity of negative coefficients for operator splitting schemes of order higher than two. Appl. Numer. Math. 54 (2005) 23–37.
- M. Caliari, S. Zuccher. A fast time splitting finite difference approach to Gross–Pitaevskii equations. Commun. Comput. Phys. 29 (2021) 1336–1364.
- S. Chin. Symplectic integrators from composite operator factorizations. Phys. Lett. A 226 (1997) 344–348.
- I. Danaila, B. Protas. Computation of ground states of the Gross–Pitaevskii functional via Riemannian optimization. SIAM J. Sci. Comput. 39 (2017) 1102–1129.
- L. Einkemmer, A. Ostermann. Overcoming order reduction in diffusion-reaction splitting. Part 2: Oblique boundary conditions. SIAM J. Sci. Comput. 38/6 (2016) A3741–A3757.
- D. Goldman, T. Kaper. n𝑛nitalic_nth-order operator splitting schemes and nonreversible systems. SIAM J. Numer. Anal. 33 (1996) 349–367.
- C. González, M. Thalhammer. A second-order Magnus-type integrator for quasi-linear parabolic problems. Math. Comp. 76/257 (2007) 205–231.
- F. Goth. Higher order auxiliary field quantum Monte Carlo methods. J. Phys. Conf. Ser. 2207 (2022) 012029.
- E. Hansen, A. Ostermann. Exponential splitting for unbounded operators. Math. Comp. 78/267 (2009) 1485–1496.
- E. Hansen, A. Ostermann. High order splitting methods for analytic semigroups exist. BIT 49 (2009) 527–542.
- T. Jahnke, C. Lubich. Error bounds for exponential operator splittings. BIT Numer. Math. 40/4 (2000) 735–744.
- H. Kleinert. Gauge Fields in Condensed Matter. World Scientific, 1989.
- E. Kieri. Stiff convergence of force-gradient operator splitting methods. Appl. Numer. Math. 94 (2015) 33–45.
- A. Lukassen, M. Kiehl. Operator splitting for chemical reaction systems with fast chemistry. J. Comput. Appl. Math. 344 (2019) 495–511.
- R. McLachlan, G. Quispel. Splitting methods. Acta Numerica 11 (2002) 341–434.
- A. Messiah. Quantum Mechanics. Dover, 1999.
- A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer, 1983.
- Q. Sheng. Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9 (1989) 199–212.
- M. Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32 (1991) 400–407.
- M. Thalhammer. High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46 (2008) 2022–2038.
- M. Thalhammer. Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258.
- H. Yoshida. Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268.