# American Institute of Mathematical Sciences

April  2021, 14(2): 303-321. doi: 10.3934/krm.2021006

## Shadow Lagrangian dynamics for superfluidity

 1 Department of Mathematics, Ruhr-University Bochum, DE-44801 Bochum, Germany 2 Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden 3 Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4 Department of Information Technology, Uppsala University, SE-751 05 Uppsala, Sweden

* Corresponding author

Received  June 2020 Revised  December 2020 Published  April 2021 Early access  January 2021

Fund Project: P. Henning acknowledge the support by the Swedish Research Council (grant 2016-03339) and the Göran Gustafsson foundation and A. M. N. Niklasson is most grateful for the hospitality and pleasant environment at the Division of Scientific Computing at the department of Information Technology at Uppsala University, where he stayed during his participation of the development of this work. This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences (FWP LANLE8AN) and by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy Contract No. 892333218NCA000001. The manuscript has the LA-UR number 19-32663

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

Citation: Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic & Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006
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##### References:
Model problem 1. Left: ground state density $|u_0|^2$ at $t = 0$. Middle: exact density $|u(t)|^2$ at time $t = 2$ Right: exact density at final computing time $t = 4$
Model Problem 1. Comparison of accuracies for the Dissipative Shadow Lagrangian Method with different dissipation orders $K$. Notably, the accuracies for $K$ = 3, 4, 5, 6 are extremely close to each other
Model Problem 1. Left: Comparison of the ${H^1}$-errors for DS-K$5$, the classical Crank-Nicolson scheme and the Besserelaxation scheme. Right: comparison of the exact ${H^1}$-error for DS-K$5$ with the estimated error using the consistency indicator $\left|E\left(\psi^{N}\right)-E\left(\phi^{N}\right)\right|$
Model Problem 1. Results obtained for DS-K$5$ and three different step sizes $\tau$. Left: Variations of the energy E (cf. $(13)$) over time. Right: Variations of the mass $m\left(\psi^{n}\right) = \left\|\psi^{n}\right\|_{L^{2}(U)}$ over time
Model problem 2. Left: potential $V_0$ as given by $(14)$. Middle: ground state density $|u_0|^2$, i.e., $|u(t)|^2$ at $t = 0$. Right: exact density $|u(t)|^2$ at final computing time $t = 1$
Model Problem 2. Comparison of the accuracies for DS-K$5$, the classical Crank-Nicolson scheme and the Besse-relaxation scheme. Left: comparison of ${L^2}$-errors. Right: comparison of ${H^1}$- errors with visibly reduced convergence rates
Model Problem 2. Left: Consistency error $\psi^{N}-\phi^{N}$ in the ${L^2}$- and the ${H^1}$-norm for various step sizes $\tau$. Right: Comparison of the exact ${H^1}$-error for DS-K$5$ with the estimated error using the consistency indicator $\left|E\left(\psi^{N}\right)-E\left(\phi^{N}\right)\right|$
Model Problem 2. Results obtained for DS-K$5$ and three different step sizes $\tau$. Left: Variations of the energy E (cf. $(15)$) over time. Right: Variations of the mass $m\left(\psi^{n}\right) = \left\|\psi^{n}\right\|_{L^{2}(U)}$ over time
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