FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

Abstract

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents). Program summaryProgram Title: FELIX-2.0Program Files doi:http://dx.doi.org/10.17632/t8b4h9g88r.1Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Computer Physics Communications 200, 350–363 (2016)Does the new version supersede the previous version?: YesReasons for the new version: FELIX-2.0 extends the physics capabilities of the previous version, since it includes the option to have a metric term in the collective Schrödinger equation. The computational efficiency of the code is considerably increased thanks to the implementation of several new numerical methods. This version also provides a more flexible and robust workflow to perform calculation of fission fragment distributions more efficiently.Summary of revisions: The new version includes the ability to use a metric term in the TDGCM+GOA equation. Numerical integration methods (both spatial and time integration) have been updated to more efficient schemes: matrix elements are now computed with Gauss–Legendre or Gauss–Legendre–Lobatto quadratures, the time propagation of the wave packet is computed with the Krylov approximation of the propagator instead of the previous Crank–Nicolson scheme. A new type of finite element based on n-dimensional orthotope is implemented, which enables a spectral element scheme for spatial discretization. Finally, the workflow and inputs/outputs of the code were entirely redesigned to provide more flexibility and be more user-friendly.Nature of problem: The Gaussian overlap approximation to the time-dependent generator coordinate method [1,2] yields a local, time-dependent Schrödinger equation in a small, multi-dimensional collective space. Its solution provides the time-evolution of the collective probability amplitude. For applications to nuclear fission, scission configurations are defined by a hyper-surface in this collective space referred to as the frontier. Distributions of nuclear observables such as charge or mass distributions of the fission fragments are extracted from the probability for the system to pass through any given section of the frontier. This probability is computed by first solving the evolution equation up to times greater than 10−20s and then integrating the flux of probability through the frontier over the entire timerange.Solution method: FELIX-2.0 solves the time-dependent Schrödinger equation by first discretizing the N-dimensional collective space with the continuous Galerkin Finite Element Method. This produces a large set of coupled, time-dependent Schrödinger equations characterized by the sparse overlap and Hamiltonian matrices. The solution is evolved in small time steps by applying an explicit and unitary propagator built as a Krylov approximation [3] of the exponential of the Hamiltonian.Additional comments including Restrictions and Unusual features: Although the implementation of the program gives it the ability to solve the TDGCM+GOA equation in a generic N-dimensional collective space, it has only been tested on 1-, 2- and 3-dimensional meshes.[1] J.J. Griffin, J. A. Wheeler, Collective Motions in Nuclei by the Method of Generator Coordinates, Phys. Rev. C 108, 311-327 (1975)[2] P. G. Reinhard, K. Goeke, The Generator-Coordinate Method and Quantized Collective Motion in Nuclear Systems, Rep. Prog. Phys. 50, 1-64 (1987)[3] Y. Saad, Iterative Methods for Sparse Linear Systems: Second Edition, SIAM, 2003