| 1. |
- Rothkopf, Alexander, et al.
(författare)
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A new variational discretization technique for initial value problems bypassing governing equations
- 2023
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 477
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by the fact that both the classical and quantum description of nature rest on causality and a variational principle, we develop a novel and highly versatile discretization prescription for classical initial value problems (IVPs). It is based on an optimization (action) functional with doubled degrees of freedom, which is discretized using a single regularized summation-by-parts (SBP) operator. Formulated as optimization task it allows us to obtain classical trajectories without the need to derive an equation of motion. The novel regularization we develop in this context is inspired by the weak imposition of initial data, often deployed in the modern treatment of IVPs and is implemented using affine coordinates. We demonstrate numerically the stability, accuracy and convergence properties of our approach in systems with classical equations of motion featuring both first and second order derivatives in time.
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| 2. |
- Rothkopf, Alexander, et al.
(författare)
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A symmetry and Noether charge preserving discretization of initial value problems
- 2024
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Ingår i: Journal of Computational Physics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9991 .- 1090-2716. ; 498
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Tidskriftsartikel (refereegranskat)abstract
- Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics along a world-line parameter, instead of physical time directly, we retain manifest translation symmetry and conservation of the associated continuum Noether charge. A non-equidistant time discretization emerges dynamically, realizing a form of automatic adaptive mesh refinement (AMR), guided by the system symmetries. Using appropriately regularized summation by parts finite difference operators, the continuum Noether charge, defined via the Killing vector associated with translation symmetry, is shown to be exactly preserved in the interior of the simulated time interval. The convergence properties of the approach are demonstrated with two explicit examples.
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| 3. |
- Rothkopf, Alexander, et al.
(författare)
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The crucial role of Lagrange multipliers in a space-time symmetry preserving discretization scheme for IVPs
- 2024
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 511
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Tidskriftsartikel (refereegranskat)abstract
- In a recently developed variational discretization scheme for second order initial value problems, it was shown that the Noether charge associated with time translation symmetry is exactly preserved in the interior of the simulated domain. The obtained solution also fulfils the naively discretized equations of motions inside the domain, except for the last two grid points. Here we provide an explanation for the deviations at the boundary as stemming from the effect of Lagrange multipliers used to implement initial and connecting conditions. We show explicitly that the Noether charge including the boundary corrections is exactly preserved at its continuum value over the whole simulation domain, including the boundary points.
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| 4. |
- Ålund, Oskar, 1987-, et al.
(författare)
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Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator
- 2021
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 425
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Tidskriftsartikel (refereegranskat)abstract
- We develop a novel numerical scheme for the simulation of dissipative quantum dynamics, following from two-body Lindblad master equations. It exactly preserves the trace of the density matrix and shows only mild deviations from hermiticity and positivity, which are the defining properties of the continuum Lindblad dynamics. The central ingredient is a new spatial difference operator, which not only fulfills the summation by parts (SBP) property, but also implements a continuum reparametrization property. Using the time evolution of a heavy-quark anti-quark bound state in a hot thermal medium as an explicit example, we show how the reparametrization neutral summation-by-parts (RN-SBP) operator enables an accurate simulation of the full dissipative dynamics of this open quantum system.
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