TY - JOUR
T1 - A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems
AU - Athanassoulis, Agissilaos
AU - Katsaounis, Theodoros
AU - Kyza, Irene
AU - Metcalfe, Stephen
N1 - The authors acknowledge the support from The Carnegie Trust Research Incentive Grant RIG008215 . I.K. would also like to acknowledge the support from London Mathematical Society through an Emmy Noether Fellowship . In addition, Th. K. and I.K. thank the Edinburgh Mathematical Society for the Covid Recovery Fund that allowed for the completion and the submission of this paper.
PY - 2023/10/1
Y1 - 2023/10/1
N2 - We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10,11,34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.
AB - We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schrödinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the nonlinearity is handled by means of a relaxation approach in the spirit of [10,11,34] for the nonlinear Schrödinger equation. For the spatial discretisation we use the standard conforming finite element scheme. The resulting scheme is explicit with respect to the nonlinearity, i.e. it requires the solution of a linear system for each time-step, and satisfies discrete versions of the system's mass conservation and energy balance laws for constant meshes. The scheme is seen to be second order in time. We conclude by presenting some numerical experiments, including an example from cosmology and an example with variable time-steps which demonstrate the effectiveness and robustness of the new scheme.
KW - Energy preserving scheme
KW - Finite element method
KW - Relaxation scheme in time
KW - Schrödinger-Poisson system
U2 - 10.1016/j.jcp.2023.112307
DO - 10.1016/j.jcp.2023.112307
M3 - Article
AN - SCOPUS:85165205580
SN - 0021-9991
VL - 490
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 112307
ER -