caa a22 4500 60549715X CHVBK 20210128100540.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s10543-014-0510-4 doi (NATIONALLICENCE)springer-10.1007/s10543-014-0510-4 Gosse Laurent Istituto per le Applicazioni del Calcolo, Via dei Taurini, 19, 00185, Rome, Italy aut A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation [Elektronische Daten] [Laurent Gosse] The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the $$L^2$$ L 2 norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of $$c$$ c , the speed of light. Moreover, when $$c$$ c diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a "diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that $$c \Delta x\rightarrow 0$$ c Δ x → 0 ). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed. Springer Science+Business Media Dordrecht, 2014 Dirac equation nationallicence One-dimensional relativistic quantum mechanics nationallicence Asymptotic-preserving and well-balanced numerical methods nationallicence BIT Numerical Mathematics Springer Netherlands 55/2(2015-06-01), 433-458 0006-3835 55:2<433 2015 55 10543 https://doi.org/10.1007/s10543-014-0510-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10543-014-0510-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Gosse Laurent Istituto per le Applicazioni del Calcolo, Via dei Taurini, 19, 00185, Rome, Italy aut NATIONALLICENCE 773 0- BIT Numerical Mathematics Springer Netherlands 55/2(2015-06-01), 433-458 0006-3835 55:2<433 2015 55 10543